Petru Mironescu May
email: mironescu@math.univ-lyon1.fr

Sobolev maps to manifolds

Keywords: Motivation. Program. Preliminary remarks Fractional Sobolev spaces, manifold constraint, density, trace theory, covering spaces, li ting, singularities, Jacobian. MSC classification. B, E

teaching and research institutions in France or abroad, or from public or private research centers.

Before proceeding further, let us note that a common feature of the above proofs is that the presence of topological invariants prevents the existence of extensions, or strongly approximating sequences, or other classical properties of scalar Sobolev spaces.

We now present a research program, in part initially developed by Bethuel in his groundbreaking contribution [ ], motivated by the pathologies exhibited in Proposition . .

General program

Strong density problems i) Characterize W s,p (Ω; N ) having the strong density property (C ∞ (Ω; N ) is strongly dense in W s,p (Ω; N )). ii) If the density property fails, find a class R of maps u enquoteas smooth as possible dense in W s,p (Ω; N ).

iii) If the density property fails, characterize the closure of C ∞ (Ω; N ) in W s,p (Ω; N ).

(Sequential) Weak density problems i) Characterize W s,p (Ω; N ) having the (sequentially) weak density property (C ∞ (Ω; N ) is (sequentially) weakly dense in W s,p (Ω; N )). ii) If the (sequentially) weak density property fails, characterize the (sequentially) weak closure of C ∞ (Ω; N ) in W s,p (Ω; N ).

Extension problems Here, we assume that s is not an integer. (We could also let s be an integer when p = 2.)

i) Characterize W s,p (Ω; N ) having the extension property: ∀ u ∈ W s,p (Ω; N ), ∃ U ∈ W s+1/p,p (Ω × (0, 1); N ) sucht that tr U = u.

ii) If the extension property fails, characterize tr W s+1/p,p (Ω × (0, 1); N ).

Li ting problems Let π : E → N be a non-trivial (locally isometric) covering map, with E a smooth embedded manifold.

i) Characterize W s,p (Ω; N ) having the li ting property, in the sense that for every u ∈ W s,p (Ω; N ) there exists some ϕ ∈ W s,p (Ω; E ) such that u = π • ϕ. ii) If the li ting property fails, characterize π • W s,p (Ω; E ).

In full generality, this program is still partly open (especially for the weak density problems, which are the new frontier). In what follows, I will present some of the main results in these directions, some basic tools and elements of proofs, and indicate additional results that are beyond the scope of these notes. Before proceeding, let us discard two cases: (i) the easy case where W s,p → C 0 (i. e., when sp > N or s = N and p = 1); (ii) the relatively easy case where sp = N .

Proposition . . Assume that W s,p → C 0 . Then W s,p (Ω; N ) has the strong (and thus weak) density property, the extension property (provided s is not an integer), and the li ting property.

Proof. The proofs of all the properties are similar: they rely on smoothing and nearest point projection on N . We detail the extension property. Let ρ ∈ C ∞ c (B 1 (0); R + ) be a standard mollifier, and set V (x, ε) := u * ρ ε (x), ∀ ε > 0, ∀ x ∈ Ω ε := {x ∈ Ω; d(x, ∂Ω) < ε}.

( . )

By standard trace theory (see, e. g., [ , Proof of Lemma . ] and Section . ), we have V ∈ W s+1/p,p on its domain. Let δ > 0 be such that the nearest point projection Π on N is smooth on the set {y ∈ R ; d(y, N ) ĺ δ}. Let ε 0 be such that

[0 < ε < ε 0 , x ∈ Ω ε ] =⇒ d(V (x, ε), N ) ĺ δ.
( . )

(The existence of ε 0 follows from the embedding W s,p → C 0 and the definition of V .) Set

T (x, ε) := Π • V (x, ε), ∀ 0 < ε < ε 0 , ∀ x ∈ Ω ε . ( . )
Then T is clearly N -valued and belongs to W s+1/p,p (Theorems . and . ). Let us next note that T is defined on

W := {(x, ε); 0 < ε < ε 0 , x ∈ Ω ε }.
( . )

Picking a di feomorphism Ψ : Ω × [0, 1] → W such that Ψ(x, 0) = (x, 0), ∀ x ∈ Ω, Ψ(Ω ε × {ε}) = Ω × {ε}, ∀ 0 ĺ ε ĺ ε 0 , we find that U := T • Ψ belongs to W s+1/p,p (Ω × (0, 1); N ). Finally, since U (x, ε) = Π • V • Ψ(x, ε) → Π • u • Ψ(x, 0) = u(x) as ε → 0, ∀ x ∈ Ω, we find that U has all the required properties.

QED

The limiting case sp = N is slightly more involved, and requires additional ingredients: the embedding W s,p → VMO (see Theorem . ) combined with a remarkable property of smoothing of VMO maps, made popular by Brezis and Nirenberg [ ] (see Lemma . below, with roots in Schoen and Uhlenbeck [ ] and Boutet de Monvel and Gabber [ ]).

Proposition . . Assume that sp = N . Then W s,p (Ω; N ) has the strong (and thus weak) density property and the extension property (provided s is not an integer).

Proof. We may assume that p > 1, since for p = 1 we have W N,1 → C 0 . We consider only the extension property. The proof is similar to the previous one. The main novelty stems in the proof of the existence of ε 0 satisfying ( . ) (see Lemma . ), since one cannot invoke anymore the continuity of u. Granted the existence of ε 0 , we construct U as above. To see that tr U = u, we argue as follows. (i) We clearly have tr V • Ψ = u. (Start by considering a smooth u, then pass to the limits, using trace theory.) (ii) Extend Π to a smooth compactly supported map, still denoted Π. By trace theory and Theorems . and . , for every map Y ∈ W s+1/p,p (Ω × (0, 1); R ), we have tr Π • Y = Π • (tr Y ). Applying (ii) to Y = V • Ψ and using (i), we find that, for our specific u, we have indeed tr U = u, as claimed. ( . )

Proof. For x ∈ Ω ε , we have

Bε(x)
|u(y) -u * ρ ε (x)| dy ĺ ˆBε(x) Bε(x) |u(y) -u(z)| |ρ ε (x -z)| dydz

ĺ ω N ρ L ∞ Bε(x) Bε(x)
|u(y) -u(z)| dydz.

We find that there exists some y 0 ∈ B ε (x) such that

|u(y 0 ) -u * ρ ε (x)| ĺ ω N ρ L ∞ Bε(x) Bε(x)
|u(y) -u(z)| dydz.

For such a y 0 , we have

d(u * ρ ε (x), F ) ĺ |u ε (x) -u(y 0 )| ĺ ω N ρ L ∞ Bε(x) Bε(x)
|u(y) -u(z)| dydz.

( . )

We conclude using the definition of VMO.

QED

In view of the above, in what follows, we assume, unless specified otherwise, that sp < N .

( . )

Also, in order to simplify the statements, in what follows, we assume, unless specified otherwise, that N ľ 2.

( . )

Lecture # . Li ting

Recall that the implicit assumptions in this section are N ľ 2 and sp < N .

Let π ∈ C ∞ (E , N ) be a Riemannian covering. We assume that: (i) E is connected, embedded into some R m ; (ii) π is locally isometric and non-trivial (i. e., π -1 (z) contains at least two points, ∀ z ∈ N ). A special important case is the one of the universal covering of a non-simply connected manifold N . Here are three prototypical examples.

. N = S 1 , E = R, π(t) = e ıt .

. N = RP k , E = S k (with k ľ 2), π(t) = {t, -t}.

. N = S 1 , E = S 1 (viewed as subsets of C), π(t) = t k , with k ∈ Z, |k|ľ 2.

The last two examples belong to the compact case where E is compact, while the first one belongs to the non-compact case (E is non-compact).

We next discuss the seminorm we consider on W s,p (Ω; E ) when 0 < s < 1. Set

|ϕ| p W s,p := ˆΩ ˆΩ [d E (ϕ(x), ϕ(y))] p |x -y| N +sp dxdy,
where d E is the geodesic distance on E . When E is compact, the above seminorm is equivalent to the one obtained by taking the Euclidean distance |ϕ(x) -ϕ(y)| in R m . This need not be the case in general.

We now present an important condition devised by Detaille [ ].

There exists some Φ ∈ C ∞ (R m , L (R , R m )) with bounded derivatives such that

Φ(t)((d t π)(τ )) = τ, ∀ t ∈ E , ∀ τ ∈ T t (E ). ( . )
This condition requires the global existence of a "controlled" le t-inverse of the isometry d t π : T t (E ) → T π(t) (N ). An explicit construction shows that this condition is automatically satisfied in the compact case [ ], but a counterexample in [ ] shows that it may not be satisfied in the non-compact case. Intuitively, ( . ) requires that the embedding of E "does not swirl too much".

Of importance for us is that this condition is satisfied by the universal covering π : R → S 1 (take Φ(t)(x 1 , x 2 ) := (-sin t)x 1 + (cos t)x 2 , ∀ (x 1 , x 2 ) ∈ R 2 ).

We have the following results (see [ ] for the universal covering of S 1 , Bethuel and Chiron [ ] for the non-compact case and partial results in the compact case, [ ] for the full result when 0 < s < 1, and Detaille [ ] for the role of the condition ( . )).

Theorem . . Let Ω = (0, 1) N . Assume that s ľ 1.

a) The li ting property fails when 1 ĺ sp < 2. b) In the non-compact case, when s > 1 further assume that ( . ) holds. The li ting property holds when sp ľ 2.

Theorem . . Let Ω = (0, 1) N . Assume that 0 < s < 1.

a) The li ting property holds when sp < 1.

b) The li ting property fails when 1 ĺ sp < 2.

c) In the non-compact case, the li ting property fails when 1 ĺ sp < N .

d) In the compact case, the li ting property holds when sp ľ 2.

Proofs.

Step . The li ting property fails when 1 ĺ sp < 2. Fix some point z 0 ∈ N . Assume, for simplicity, that z 0 = 0.

We first explain a gluing construction, valid for each integer 2 ĺ k ĺ N . (In our specific case, we will take k = 2.) Consider, for some 0 < a < 1, the cone C := {x = (x 1 , x ) ∈ R k ; |x |ĺ ax 1 }. Consider a sequence of maps u j = u j (x ), smooth in R k \{0}, such that u j (x ) = 0 if x ∈ C and u j ∈ W s,p (B 1 (0)). Write a point in R N in the form x = (x , y) ∈ R k × R N -k . We will define inductively points b j ∈ (0, 1) k and maps w j , such that:

(i) The truncated cones (C + b j ) ∩ [0, 1] k are mutually disjoint.

(ii) {1} × (0, 1) k-1 ⊂ ∪ iĺj (C + b i ), ∀ j.

(iii) w j (x) = u j (x -b j ), ∀ j, ∀ x ∈ Ω.

(iv) ||w j || W s,p ĺ 2 -j , ∀ j ľ 2.

Let b 1 = 0. Assume that we have chosen b 1 , . . . , b j . Pick some point (1, x ) ∈ {1} × (0, 1) k-1 \ ∪ iĺj (C + b i ). Then, for su ficiently small ε, the vertex b j := (1 -ε, x ) has all the required properties.

Moreover, clearly, by construction, we also have

(v) Ω \ ∪ jľ1 [(C + b j ) × (0, 1) N -k ] is connected.
(vi) The map w := w j , in (C + b j ) × (0, 1) N -k 0, in Ω \ ∪ jľ1 [(C + b j ) × (0, 1) N -k ] belongs to W s,p and is smooth in

Ω \ ∪ jľ1 [{b j } × (0, 1) N -k ].
We now specialize to our situation. Recall that we have assumed that z 0 = 0. Let (t j ) jľ1 be an enumeration (possibly with repetitions) of π -1 (0). Let 0 < b < π/4. By Lemma . , there exists some some v j ∈ C ∞ (R; N ) such that v j (θ) = 0, ∀ j, ∀ |θ|ľ b, and there exists no li ting

ζ ∈ C([-b, b]; E ) of v j on [-b, b] such that ζ(-b) = ζ(b) = t j . Set u j (re ıθ ) := v j (θ), ∀ r > 0, ∀ θ ∈ R.
Then, clearly, u j satisfies the assumptions at the beginning of this step (the property u j ∈ W s,p (B 1 (0)) following from Lemma . c)). Let w be as above. We claim that w has no li ting ϕ ∈ W s,p . Argue by contradiction. By Lemma . , ϕ is continuous in the connected set U := Ω \ ∪ jľ1 [{b j } × (0, 1) N -2 ]. Since w = 0 in U , there exists some j such that ϕ = t j in U . By continuity, ϕ = t j on the set [(∂C + b j ) ∩ Ω] \ [{b j } × (0, 1) N -2 ]. In particular, for small ε > 0, ϕ(b j + εe ±ıb ) = t j . Going back to the definition of w j , we find that v j has, on [-b, b] For pedagogical reasons, the case s = 1 is split into two sub-cases.

Step . The li ting property holds when s = 1 and p ľ 2: the compact case. Let first u be in the class R in ( . ), with := N -p -1 ĺ N -3. With A as in ( . ), the open set U := Ω \ A is simply connected (Lemma . ). Therefore, u has a smooth li ting ϕ in U . Since |∇ϕ|= |∇u| pointwise, we find that ϕ ∈ W 1,p (Ω; E ) and ||∇ϕ|| p = ||∇u|| p . Moreover, E being compact, we have

||ϕ|| p ĺ C := max{|t|; t ∈ E }. Let now u ∈ W 1,p (Ω; N ). Consider a sequence (u j ) ⊂ R such that u j → u in W 1,p (cf Theorem . ). The corresponding sequence (ϕ j ) of li tings is bounded in W 1,p (Ω; E ). If ϕ ∈ W 1,p (Ω; E ) is such that, up to a subsequence, ϕ j ϕ, then ϕ ∈ W 1,p
and ϕ is a li ting of u.

Step . The li ting property holds when s = 1 and p ľ 2: the non-compact case. The additional issue is passing to the weak limit the ϕ j 's, since boundedness in L p is not guaranteed anymore. This is achieved by induction on the space dimension, via the following result. Let N ľ 2 and let (u j ) ⊂ W 1,p (Ω; N ) be a convergent sequence. Then there exists a bounded sequence (ϕ j ) ⊂ W 1,p (Ω; E ) such that π • ϕ j = u j , ∀ j. (Actually, this result also holds when E is compact, but the proof in Step allows us to avoid its use.)

Step . . The induction process. Assume that the above property holds for N -1. Let u ∈ W 1,p (Ω; N ). Consider a sequence (u j ) ⊂ R such that u j → u in W 1,p . Let A j be the corresponding singular sets as in ( . ). Possibly a ter passing to a subsequence, for a.e. θ ∈ (0, 1), the partial map u j,θ := u(•, θ) belongs to W 1,p ((0, 1) N -1 ; N ), the set A j,θ := {x ∈ (0, 1) N -1 ; (x , θ) ∈ A j } is a finite union of N -p -2 planes (this condition being empty when N < p + 2) and u j,θ → u(•, θ) in W 1,p . Pick such θ. Let ψ j ∈ W 1,p ((0, 1) N -1 ; E ) be a bounded sequence of li tings of u j,θ (cf the induction assumption). By Lemma . , ψ j is smooth in (0, 1) N -1 \ A j,θ . Fix some x 0 ∈ (0, 1) N -1 \ A j,θ and let ϕ j be the smooth li ting of u j in Ω \ A such that ϕ j (x 0 , θ) = ψ j (x 0 ). (The existence of ϕ j follows from Lemma . .) Since |∇ϕ j |= |∇u j | in Ω \ A, we find that ϕ j ∈ W 1,p (Ω). On the other hand, we clearly have tr (0,1) N -1 ×{θ} ϕ j = ψ j . By the induction assumption and the standard inequality

||ϕ j || p À ||ψ j || p + ||∇ϕ j || p ,
we find that (ϕ j ) is bounded in W 1,p . As in Step , we obtain that u has a li ting ϕ ∈ W 1,p .

Step . . The case N = 2. Let (u j ) ⊂ W 1,p ((0, 1) 2 ; N ) be a convergent sequence. We let, as in the proof of Proposition . , V j (x, ε) := u j * ρ ε (x). We claim that there exists an ε 0 (depending on the sequence (u j )) such that ( . ) holds uniformly in j. When p > 2, this is clear, by Morrey's embedding W 1,p → C 0,1-2/p . When p = 2, the claim follows from an inspection of the proof of Lemma . . Extend u j by reflection across ∂Ω. Continuing the calculation ( . ), we find, for small ε, that

d(u j * ρ ε (x), F ) À Bε(x) Bε(x) |u j (y) -u j (z)| dydz Àε -2N ˆB2ε (0) ˆBε(x) |u j (z + h) -u j (z)| dzdh ĺε -2N ˆB2ε (0) ˆB3ε (x) |h| |∇u j (y)| dydh Àε 1-N ˆB3ε (x) |∇u j (y)| dy À ε 1-N +N (p-1)/p ||∇u j || L p (B 3ε (x)) .
Recalling that, in our case, N = 2 and p = 2, the claim follows from the above calculation, the assumption that (∇u j ) converges in L 2 , and Lebesgue's lemma.

Associate now to u j the map T j as in ( . ). Fix some M < ∞ such that π(B M (0) ∩ E ) = N . Fix some x 0 ∈ Ω and consider some t j ∈ E such that |t j |ĺ M and π(t j ) = T j (x 0 , ε 0 ). By construction, T j is smooth in Ω × (0, ε 0 ), and Lipschitz (with a Lipschitz constant independent of j) on Ω × {ε 0 }. In addition, we have

|∇ x T j (•, ε)| p À ||∇u j || p ( . )
(with constants independent of j and 0 < ε ĺ ε 0 ) and, by standard trace theory,

||∇T j || L p (Ω×(0,ε 0 )) À |u j | W 1-1/p,p À ||∇u j || p . ( . )
Let ζ j be the smooth li ting of T j on Ω×(0, ε 0 ) with ζ j (x 0 , ε 0 ) = t j . By the above, ζ j (•, ε 0 ) is Lipschitz, with controlled Lipschitz constant, and uniformly bounded at x 0 , and thus ||ζ j (•, ε 0 )|| p À 1. This, together with the L p bound ( . ), implies that

||ζ j (•, ε)|| p À ||∇u j || p + 1, ∀ j, ∀ 0 < ε < ε 0 .
Combining this with ( . ), we find that ζ j (•, ε) is uniformly bounded in W 1,p (Ω). We obtain the desired conclusion by letting ϕ j be any weak limit of a sequence of the form (ζ j (•, ε k )), with ε k → 0.

Step . The li ting property holds when s > 1 and sp ľ 2. Let u ∈ W s,p (Ω; N ) ⊂ W 1,sp (Ω; N ) (Corollary . ). By Step , there exists some ϕ ∈ W 1,sp (Ω; E ) such that u = π • ϕ. We find that d x u = d ϕ(x) πd x u, fo a.e. x ∈ Ω, and thus

Dϕ = Φ • ϕ Du, ( .
)
where Φ is as in ( . ). We complete this step by proving that

[s ľ 1, ϕ ∈ W 1,sp , u ∈ W s,p ∩ L ∞ , ( . ) holds] =⇒ ϕ ∈ W s,p . ( . )
Step . . Proof of ( . ) when s is an integer. The proof is by induction on s, the case s = 1 being clear.

The key fact is that ( . ) allows to express D s ϕ in terms of Dϕ, . . . , D s-1 ϕ. Let, e.g,

s = 2. We claim that, if u ∈ W 2,p ∩ L ∞ and v ∈ W 1,2p ∩ L ∞ , then v Du ∈ W 1,p . When u, v are smooth, this follows from the Gagliardo-Nirenberg embedding W 2,∞ ∩ L ∞ → W 1,2p
combined with the identity (with loose notation) D(vDu) = DuDv + vD 2 u. The general case follows by a standard limiting procedure. Combining this with ( . ), we find that ( . ) holds when s = 2. Moreover, we find that

|D 2 ϕ|À |D 2 u|+|Du| 2 .
The general case is obtained by an obvious argument. Let u ∈ W s,p ∩ L ∞ . By di ferentiating ( . ) (s -1)-times, we find that

|D s ϕ|À jľ1 1 +•••+ j =s |D 1 u|• • • |D j u|.
( . )

(again, first formally, then using a limiting procedure). In the process, we use the assumption ( . ). We conclude using the fact that, by the Gagliardo-Nirenberg embeddings

W s,p ∩ L ∞ → W k,sp/k , ∀ 1 ĺ k ĺ s -1, the right-hand side of ( . ) is in L p .
Step . . Proof of ( . ) when s is not an integer. Write s = k + σ, with 0 < σ < 1. By Step . and the Gagliardo-Nirenberg embedding W s,p ∩ L ∞ → W k,sp/k , we have ϕ ∈ W k,sp/k . By Theorem . and ( . ), we have J • ϕ ∈ W k,sp/k ∩ L ∞ , while, clearly, Du ∈ W s-1,p ∩ L sp . By ( . ) and Lemma . (with f := J • ϕ, g := Du, s -1 := k, s 2 := s -1, p 1 := sp/k, p 2 = p, r = sp), we have Dϕ ∈ W s-1,p , whence the conclusion.

Step . The li ting property fails in the non-compact case when 0 < s < 1 and 1 ĺ sp < N . We first present the simple special case of the universal cover π : R → S 1 . Assume for simplicity that Ω = B 1 (0). Let α > 0 and set ζ(x) := |x| -α and u := e ıζ . By Lemmas . b) and . , if ) then ζ ∈ W s,p , but u ∈ W s,p . Argue by contradiction and assume that u has a W s,p -li ting ϕ.

N -sp p ĺ α < N -sp sp , ( .
Then ϕ is continuous in B 1 (0) \ {0}, and thus there exists some k ∈ Z such that ϕ = ζ + 2kπ, a contradiction with the fact that ζ ∈ W s,p . We next explain how to treat the general case. As in Step , we rely on a gluing construction. We explain here the idea and postpone the explicit construction to Section (Lemma . ). Fix some z ∈ N and let π -1 ({z}) = {t j ; j ∈ J}. We will construct points x j ∈ Ω, radii r j > 0, j ∈ J, and maps ζ j : B 2r j (x j ) → E such that:

(i) The balls B 3r j (x j ) are mutually disjoint and contained in Ω.

(ii) ζ j ∈ C ∞ (B 2r j (x j ) \ {x j }). (iii) ζ j = t j in B 2r j (x j ) \ B r j (x j ). (iv) ζ j ∈ W s,p (B 2r j (x j )). (v) The map u := π • ζ j , in B 2r j (x j ) z, in Ω \ ∪ j B 2r j (x j ) belongs to W s,p .
Granted the existence of such ζ, we argue as follows. Assume, by contradiction, that u has a li ting ϕ ∈ W s,p . By Lemma . , ϕ is continuous in the connected set

V := Ω \ ∪ j {x j }. Let U := Ω \ ∪ j B 2r j (x j ), which is a connected set contained in V . Note that u = z in U .
Let j be such that ϕ -1 (t j ) ∩ U is non-empty. By connectedness of U and continuity of ϕ in U , we have ϕ = t j in U , and, again, by continuity and connectedness, ϕ = ζ j in B 2r j (x j ) \ {x j }. This contradicts item (iv).

Step . The li ting property holds in the compact case when 0 < s < 1 and sp ľ 2. We argue by density.

It su fices to find, for u ∈ R, a li ting ϕ such that |ϕ| W s,p À |u| W s,p . (Then we may pass to weak limits.) Set

ν := N -sp -1 ĺ N -3. Let u ∈ R. Then u is smooth in U := Ω \ A,
where A is a union of ν-planes parallel to the ν-coordinate planes. By Lemma . , U is simply connected, and thus u has a smooth li ting ϕ in U . Note that the set U N := {x ∈ (0, 1) N -1 ; [{x }×(0, 1)]∩A = ∅} is a null set, and the same holds for

U j := {(x , x ) ∈ (0, 1) j × (0, 1) N -j-1 ; [{x } × (0, 1) × {x }] ∩ A = ∅}, 1 ĺ j ĺ N -1.
Consider the partial function

x N → v := u(x , x N ), x N → ψ := ϕ(x , x N ), with x ∈ U N and x N ∈ (0, 1). Let 0 < θ 1 < θ 2 < 1. Let ρ := inj(N ) > 0 (the injectivity radius of N ). If v(θ) ∈ B ρ (v(θ 1 )), ∀ θ 1 ĺ θ ĺ θ 2 , then d E (ψ(θ), ψ(θ 1 )) = d N (v(θ), v(θ 1 )), ∀ θ 1 ĺ θ ĺ θ 2 .
( . )

Combining ( . ) with the fact that ϕ is uniformly bounded (since E is compact), we find that, for every θ 1 , θ 2 , we have

d E (ψ(θ 2 ), ψ(θ 1 )) À |v| W 0,∞ ((θ 1 ,θ 2 )) := v - θ 2 θ 1 v L ∞ ((θ 1 ,θ 2 ))
.

( . )

From ( . ) and Corollary . , we obtain the linear estimate . ) and similar estimates hold for each U j .

|ψ| p W s,p ((0,1)) À |v| p W s,p ((0,1)) , ∀ x ∈ U N , (
Combining ( . ) with slicing (Theorem . ), we obtain the linear estimate |ϕ| W s,p À |u| W s,p , which allows us to complete Step .

Step . The li ting property holds when sp < 1. We argue again by density. Consider a grid of size ε with faces parallel to the coordinate hyperplanes having the origin as an edge, and N -valued maps constant on each cube of the grid. By (the proof of) Theorem . , the restrictions to Ω of such maps are dense in W s,p (Ω; N ). It su fices to obtain, for each such map, a W s,p li ting with a norm control. In order to further simplify the presentation, we assume that ε = 2 -J for some integer J. (This is not relevant for the validity of the final result.) We may now formalize our program. For k ľ 0, let P k denote the collection of dyadic cubes Q k of size 2 -k in Ω. We let F k denote the set of the (step) functions constant on each Q k . We will complete Step by proving the following: for every J ľ 0 and every u : Ω → N , u ∈ F J , there exists some li ting ϕ ∈ F J of u such that

||ϕ|| W s,p À 1 + |u| W s,p . ( . )
The construction is relatively involved. We first construct approximations of u at the larger scales 2 -k , 0 ĺ k < J, as follows. Fix once for all some point z * ∈ N . Let δ > 0 be such that the nearest point projection Π on N is well-defined and smooth in the δ-neighborhood

N δ of N . Let E k (x) := ffl Q k u, ∀ Q k ∈ P k , ∀ x ∈ Q k , and set u k (x) := Π(E k (x)), if d(E k (x), N ) ĺ δ z * , if d(E k (x), N ) > δ .
Note that E k and u k belong to F k , ∀ k.

We next construct, inductively, a li ting ϕ k of u k , 0 ĺ k ĺ J, and finally set ϕ := ϕ J . The construction goes as follows. Fix once for all some t * ∈ π -1 (z * ). Let z 0 be the value of u 0 and let ϕ 0 ∈ E be a point in π -1 ({z 0 }), nearest from t * . Inductively, given Q j+1 ∈ P j+1 , let Q j ∈ P j be such that Q j+1 ⊂ Q j . If t j is the value of ϕ j on Q j and z j+1 is the value of u j+1 on Q j+1 , then the value of ϕ j+1 on Q j+1 is a point in π -1 ({z j+1 }), nearest from t j . Clearly, ϕ k ∈ F k and π•ϕ k = u k .

In order to estimate the W s,p -norm of ϕ, we rely on the following inequalities:

d E (t, π -1 ({z})) ĺ d N (π(t), z), ∀ t ∈ E , ∀ z ∈ N , ( . ) d N (u k (x), u k-1 (x)) À f k (x) := |u(x) -E k (x)|+|u(x) -E k-1 (x)|, ∀ k ľ 1, ∀ x, ( . ) d N (u 0 (x), z * ) À f 0 (x) := |z * -E 0 (x)|+|u(x) -E 0 (x)|À 1 + |u(x) -E 0 (x)|, ∀ x. ( . )
The first property is clear, since a geodesic γ of length L from π(t) to z li ts to a curve of length L from t to some point in π -1 ({z}). For the second one, if both E k (x) and E k-1 (x) are in N δ , then ( . ) holds, since

d N (u k (x), u k-1 (x)) À|u k (x) -u k-1 (x)|= |Π(E k (x)) -Π(E k-1 (x))|ĺ |E k (x) -E k-1 (x)| ĺ|u(x) -E k (x)|+|u(x) -E k-1 (x)|
(the first inequality following from the fact that the geodesic distance and the Euclidean distance are equivalent on N ).

On the other hand, if, say, E k (x) ∈ N δ , then |u(x) -E k (x)|ľ δ, so that the right-hand side of ( . ) is at least δ, while the le t-hand side of ( . ) is dominated by sup{d N (z, w); z, w ∈ N } < ∞. Thus ( . ) holds in all cases.

The proof of ( . ) is similar to the one of ( . ). Going back to the construction of ϕ, let us note that f k ∈ E k , ∀ k, and that, by combining ( . )-( . ) with the construction of the ϕ k 's we have, for every j, every Q j ∈ P j , and every x, y ∈ Q j :

d E (ϕ(x), ϕ(y)) =d E (ϕ J (x), ϕ J (y)) ĺ 1ĺkĺJ [d E (ϕ k (x), ϕ k-1 (x)) + d E (ϕ k (y), ϕ k-1 (y))] = j<kĺJ [d E (ϕ k (x), ϕ k-1 (x)) + d E (ϕ k (y), ϕ k-1 (y))] := j<kĺJ [g k (x) + g k (y)], ( . )
where

g k (x) := d E (ϕ k (x), ϕ k-1 (x)) À d N (u k (x), u k-1 (x)) À f k (x), ∀ k ľ 1. ( . )
Using ( . ) and ( . ), the fact that g k ∈ F k , ∀ k, the assumption sp < 1, and, successively, Lemmas . and . in Section , we find that

|ϕ| p W s,p À kľ1 2 spk ||g k || p p À kľ0 2 spk ||u -E k || p p À |u| p W s,p . ( . )
On the other hand, we have, by Hölder's inequality, the construction of the ϕ k 's, ( . )-( . ), ( . ), and Lemma . ,

||ϕ|| p p ĺ ˆ||ϕ 0 || p + kľ0 ||ϕ k+1 -ϕ k || p ˙p À ||ϕ 0 || p p + kľ0 2 spk ||ϕ k+1 -ϕ k || p p À1 + kľ0 2 spk ||u -E k || p p À 1 + |u| p W s,p . ( . )
Step follows from ( . ) and ( . ). The proof of Theorem . is complete.

QED

We next investigate the existence of li ting in the limiting case sp = N , which was le t apart in the previous section. We also consider the case where N = 1, which is of interest here. By Theorems . and . , a W s,p -li ting does exist when s ľ 1 or, in the compact case, when N ľ 2. We may thus assume that we are in the cases uncovered by the previous results, i. e., Assume that 0 < s < 1 and, if N ľ 2, that E is non-compact.

( . )

Let k ľ 1 be the least integer such that s + k/p ľ 1. We make a second assumption.

If s + k/p > 1 and E is non-compact, then ( . ) holds.

( . )

Proposition . . Assume ( . )-( . ). Then the li ting property holds.

Proof. Let u ∈ W s,p (Ω; N ). We first construct successive extensions of u until we reach the framework of Theorem . . This goes as follows. Arguing as in the proof of Proposition . , u has a smooth extension u 1 ∈ W s+1/p,p (Ω × (0, 1); N ). Continuing inductively as above, if k ľ 1 is the least integer such that s + k/p ľ 1, the final map u k is smooth in Ω × (0, 1) k , and belongs to W s+k/p,p (Ω × (0, 1) k ; N ). Since s + k/p ľ 1 and (s + k/p)p ľ 2, we are in position to apply Theorem . (here, when s+k/p > 1 we use the assumption ( . )), and obtain that u k has a phase ϕ k ∈ W s+k/p,p (Ω × (0, 1) k ; E ). By Lemma . , ϕ k is smooth. By trace theory adapted to E -valued maps (see Lemma . and Chiron [ , Section . ]), we have the uniform estimate

||ϕ k (•, x )|| W s,p (Ω) À ||ϕ k || W s+k/p,p (Ω×(0,1) k ) , ∀ x ∈ (0, 1) k . ( . )
By ( . ), the obvious embedding W s,p (Ω; E ) → W s,p (Ω; R m ), standard trace theory, and the compactness of the embedding W s,p (Ω; R m ) → L p (Ω; R m ), we obtain the existence of a sequence x j → 0 and a ϕ ∈ W s,p (Ω; E ) such that ϕ k (•, x j ) → ϕ a.e. and ϕ is a li ting of u.

QED Finally, we prove that, in the proof of Proposition . , the assumption ( . ) is just an artefact of the proof, and can be removed using a di ferent approach, as in [ ].

Proposition . . Assume ( . ). Then the li ting property holds.

Proof. Let V , Ω ε , and T be as in ( . )-( . ), and Ψ be a di feomorphism as in the proof of Proposition . . Let ε 0 be such that ( . ) holds. In the simply connected set W given by ( . ), T has a li ting Φ such that

|∇Φ|= |∇T |À |∇V |.
( . )

In order to explain our proof, we first invoke the following local version of the theory of weighted Sobolev spaces (see [ , proof of Lemma . ] for a proof when Ω = T N ; the argument there can be adapted to any Lipschitz bounded domain)

|Φ(•, θ)| p W s,p (Ω) ĺ C(ε 0 ) ˆΩ ˆε0 0 ε p(1-s)-1 |∇Φ(x, ε)| p dεdx, ∀ 0 < θ < ε 0 . ( . )
We call the attention of the reader to the fact that, in ( . ), the | | W s,p seminorm is calculated with respect to the Euclidean distance in R m . However, the proof of ( . ) is obtained starting from

|Φ(x + h, θ) -Φ(x, θ)|ĺ|Φ(x + h, θ) -Φ(x + h/2, θ + |h|/2)| ( . ) + |Φ(x + h/2, θ + |h|/2) -Φ(x, θ)| ĺ ˆ1 0 d dτ Φ(x + h/2 + τ h/2, θ + |h|/2 -τ |h|/2) dτ + ˆ1 0 d dτ Φ(x + τ h/2, θ + τ |h|/2) dτ, ∀ (x, h) ∈ Ω × R N s. t. |h|ĺ ε 0 2 and [x, x + h] ⊂ Ω.
Clearly, we may replace, in ( . ), the Euclidean distance with the geodesic distance d E on E , and find that ( . ) still holds for the adapted W s,p -seminorm in W s,p (Ω; E ). By ( . ), ( . ), and standard inverse trace theory (see [ ]), we have

|Φ(•, θ)| p W s,p (Ω) ĺ C(ε 0 ) ˆΩ ˆε0 0 ε p(1-s)-1 |∇T (x, ε)| p dεdx À |u| s W s,p , ∀ 0 < θ < ε 0 . ( . )
Using ( . ) and a standard limiting procedure, we find that Φ(•, θ) has a weak limit ϕ ∈ W s,p (Ω; E ) as ε → 0, satisfying π • ϕ = u.

QED

Lecture # . Strong density

Recall the implicit assumption sp < N . In this section, we let Ω = (0, 1) N .

If 0 ĺ ν ĺ N -1 is an integer, let R = R ν := {u : Ω → N ; ∃ ε > 0, ∃ a finite union A of ν-planes parallel to the ν-coordinate planes, ∃ U ∈ C ∞ ([-ε, 1 + ε] N \ A; N ) such that u = U |(0,1) N and |D k U (x)|ĺ C k [d(x, A)] -k , ∀ k ľ 0}.
( . )

The importance of the class R, devised by Bethuel [ ], is illustrated by the following result (see Bethuel [ ] for s = 1, Bousquet, Ponce, and Van Scha tingen [ ] for s = 2, 3, . . ., [ ] for 0 < s < 1, and Detaille [ ] for the remaining cases).

Theorem . . Let

ν := N -sp -1. The class R ν is dense in W s,p (Ω; N ).
Theorem . is complemented by the following result (same references as above).

Theorem . . C ∞ (Ω; N ) is dense in W s,p (Ω; N ) if and only if π sp (N ) is trivial.
The full proofs of the above results require more than hundred pages. We will present here only four elements of proof:

. The necessity of the assumption that π sp (N ) is trivial in Theorem . (following essentially Schoen and Uhlenbeck [ ]).

. Approximation with homogeneous maps when 0 < s < 1 and 1 ĺ sp < N (following [ , Section ]).

. Smoothing of homogeneous maps when s = 1 (following essentially Hang and Lin [ , Sections . and ]). Proof. We actually establish the stronger result that C ∞ (Ω; N ) ∩ W s,p (Ω; N ) is not dense in W s,p (Ω; N ). For simplicity of the formulas, we work in B 1 (0) instead of (0, 1) N .

Let

k := sp ĺ N -1. Assume first that k ľ 1. Consider some v ∈ C ∞ (S k ; N ) that is not null-homotopic. Let u(x) := v((x 1 , . . . , x k+1 )/|(x 1 , . . . , x k+1 )|)
. By Lemma . c), we have u ∈ W s,p (Ω; N ). We claim that u cannot be approximated with smooth N -valued maps. Argue by contradiction and let (u j ) ⊂ C ∞ (Ω; N ) ∩ W s,p (Ω; N ) be such that u j → u in W s,p . By slicing (Corollary . ), up to a subsequence and for some 0 < r < 1/2 and x ∈ R N -k-1 such that |x |< 1/2, we have u j (•, x ) → u(•, x ) in W s,p (rS k-1 ; N ). By Corollary . , for large j, u j (•, x ) and u(•, x ) are homotopic as continuous functions from rS k-1 to N . However, on the one hand u(•, x ) is not null-homotopic (for otherwise, v would also be), while u j (•, x ) is always null-homotopic (by a homotopy argument, since u j is smooth in B 1 (0)). The contradiction achieves the proof when k ľ 1.

Assume next that k = 0, i. e., that sp < 1. Since we work with connected N 's, π 0 (N ) is trivial. For the record, let us note that, if N is not connected, then density fails. To see this, let

C 1 , . . . , C k , k ľ 2, be the components of N . Let a m ∈ C m , m = 1, 2. Consider a ball B with B ⊂ Ω. Let u = a 1 , in B a 2 , in Ω \ B . Then u ∈ W s,p (Ω; N ) (Lemma .
). We claim that u cannot be approximated with a sequence (u j ) of smooth N -valued maps. Argue by contradiction. Up to a subsequence, we have u j → u a.e., and in particular, for a given ε > 0 and large j, the sets {x ∈ Ω; |u j (x) -a m |< ε} have positive measure. If we let, in particular, ε < min{d(C i , C m ), i = m}, we find that, for large j, u j has to take values both in C 1 and C 2 . However, this cannot happen, since the image of u j is connected.

QED

. . Approximation with homogeneous maps when 0 < s < 1

In this section, | |stands for the || || ∞ norm.

We start by describing a procedure for constructing homogeneous maps on R N . Fix some ε > 0 and t ∈ R N . Consider the mesh C N = C N,t = C N,t,ε of N -dimensional cubes (with vertices parallel to the coordinate axes) of side-length 2ε having t as one of the centers. (Thus, cubes in C N are of the form t

+ 2εk + [-ε, ε] N , with k ∈ Z N .) Let C N -1 = C N -1,
t be the (N -1)-dimensional skeleton associated with this mesh, i. e., C N -1 is the union of the boundaries of the cubes in C N . Let H N be the mapping that associates with every g :

C N -1 → R its homogeneous extension (on each cube of C N ) to R . Analytically, if C is a cube in C N , of center u, then H N (g)(x) = g(u + ε(x -u)/|x -u|), ∀ x ∈ C.
In order to keep notation reasonably simple, we will identify C j with the union of its cubes, so that we write both C ∈ C j and, if x ∈ C, x ∈ C j .

We next consider a more general situation. We start by defining the lower dimensional skeletons and cubes associated with C N . This is done by backward induction:

C N -2 = C N -2,t = C N -2,t,ε is the union of the (N -2)-dimensional boundaries of the cubes in C N -1 = C N -1,t = C N -1,t,ε
, and so on. A cube in C N is any cube of the mesh C N . A cube in C N -1 is any of the 2N faces of a cube in C N . For j ĺ N -2, a cube in C j is any of the 2(j + 1) faces of any cube in C j+1 .

For g : C j → R , let H j+1 (g) be its homogeneous extension to C j+1 .

Let 0 ĺ j < N . For ε > 0 and t ∈ R N , we associate with each map f : R N → R a map

f t = f t,ε = f t,ε,j : R N → R through the formula f t = g j := H N • H N -1 • • • • • H j+1 • g; here, we set g := f | C j .
( . )

More generally, given any map g : C j → R , the map g j given by the right-hand side of ( . ) is referred to as a j-homogeneous map or the j-homogeneous extension of g.

Here is our main result in this section.

Theorem . . Let 0 ĺ j < N, 0 < s < 1, sp < j + 1, and let f ∈ W s,p (R N ; R ). Then there exist sequences ε k → 0 and (t k ) ⊂ R N such that f t k ,ε k → f in W s,p (R N ).
For the sake of simplicity, we prove Theorem . under the extra assumptions

1 < p < ∞, j ľ 1.
( . ) (Theorem . holds without these assumptions, but the treatment of the remaining cases is more involved.) Under these assumptions, we will obtain the following improvement of Theorem . .

Theorem . . Assume 1 ĺ j < N , 0 < s < 1, 1 < p < ∞, and sp < j + 1. Let f ∈ W s,p (R N ; R ). Then, for each ε ∈ R N , there exists some t ε ∈ R N such that f tε,ε → f in W s,p (R N ) as ε → 0.
Before proceeding to the proof of Theorem . , let us note the following consequence of Theorem . . Corollary . . Assume that 0 < s < 1 and sp < N . Let F ⊂ R be an arbitrary set. Let f ∈ W s,p (Ω; F ). Then there exists a sequence of j-homogeneous maps

(f k ) ⊂ W s,p (R N ; R ) such that f k → f in W s,p (Ω) and f k is F -valued in (-1, 2) N . Proof. Extend f , by reflexions, to a map r f ∈ W s,p ((-2, 3) N ; F ). Then extend r f to a map h ∈ W s,p (R N ; R ). (We do not claim that h is F -valued.) Finally, let f k := h t k ,ε k , with ε k , t k as in Theorem .
(applied to h). Then the f k 's have, for large k (and thus small ε k ), the desired properties.

QED

Proof of Theorem . . We start by introducing some useful notation. Set Q ε := [-ε, ε] N . In order to keep notation easier to follow, we will sometimes denote a point in Q ε by x N rather than x. We denote by x N -1 the radial projection (centered at 0) of x N onto the (N -1)-skeleton (thus the boundary) of Q ε ; this projection is defined except when x N = 0. With an abuse of notation, x N -1 also denotes a "generic" point of ∂Q ε . We next let x N -2 denote the radial projection of x N -1 onto the (N -2)-skeleton of Q ε . The point x N -2 is obtained as follows: if x N -1 ∈ ∂Q ε belongs to an (N -1)-dimensional face F of ∂Q ε , and if x N -1 is not the center C of F , then the radial projection (centered at C) of x N -1 on ∂F is well-defined, and yields x N -2 . By backward induction, we define x j , 0 ĺ j ĺ N -1, as the radial projection of x j+1 onto ∂Q ε ∩ C j,0 ; this is defined for all but a finite number of x j+1 's. Again, with an abuse of notation, x j is the "generic" point of ∂Q ε ∩ C j,0 . Note that x 0 is one of the vertices of Q ε . When x j is obtained starting from x N , we will denote x j as the radial projection of x N (onto ∂Q ε ∩ C j,0 ). This projection is defined except on a set of finite H N -j-1 measure.

More generally, let j < k ĺ N . We identify x k with a "generic" point of

∂Q ε ∩ C k,0 . Then x j is the projection of x k onto ∂Q ε ∩ C j,0 (except for a set of x k 's of finite H k-j-1 measure). Let k ∈ Z N and set u = t + 2εk. Then the radial projection of u + x N onto C j is u + x j . If j < k ĺ N , then for H k -a.e. x k ∈ ∂Q ε ∩ C k,0 , the projection of u + x k onto C j is u + x j .
With the above notation, formula ( . ) is equivalent to

f t (t + 2εk + x N ) = f (t + 2εk + x j ), ∀ k ∈ Z N , for H N -a.e. x N ∈ Q ε . ( . )
We now proceed to the proof of the theorem. Set

F ε (f )(t, x) := f t,ε (x), ∀ t ∈ Q ε , ∀ x ∈ R N .
Step . An L q -estimate for

F ε (f ). Let 1 ĺ q < ∞ and f ∈ L q (R N ). We claim that lim ε→0 1 ε N ˆQε ||f -f t,ε || q q dt = 0 ( . )
and

F ε (f ) q ĺ C ε N/q f q , with C independent of ε or f . ( . )
(Here, we do not require j ľ 1.)

Set Q ε (x) := x + Q ε , ∀ x ∈ R N . Using the facts that: (i) Q ε (t + 2εk)) k∈Z n is an a.e. partition of R N ; (ii) f t,ε = f t+2εk,ε for t ∈ R N and k ∈ Z N (thanks to ( .
)), and (iii) the "change of variable"

t = x + y N , ∀ t ∈ Q ε (x) (with y N ∈ Q ε ), we have 1 ε N ˆQε ||f -f t,ε || q q dt = 1 ε N ˆQε k∈Z N ˆQε(t+2εk) |f (x) -f t,ε (x)| q dxdt = 1 ε N ˆRN ˆQε(x) |f (x) -f t,ε (x)| q dtdx = 1 ε N ˆRN ˆQε ˇˇf (x) -f `x + y N -y j ˘ˇq dydx = 1 ε N ˆQε f (•) -f (• + y N -y j ) q q dy. We next note that y N ∈ Q ε =⇒ y N -y j ∈ Q ε . Therefore, 1 ε N ˆQε ||f -f t,ε || q q dt ĺ 2 N sup{||f (•) -f (• + z)|| q q ; |z|ĺ ε}. ( . )
Finally, we note that ( . ) implies both ( . ) and ( . ).

Step . A W 1,r estimate for

F ε (f ). Let 1 ĺ j ĺ N -1, 1 ĺ r < j + 1, and f ∈ W 1,r (R N ). We claim that F ε (f ) L r (Qε;W 1,r (R N )) ĺ C ε N/r f W 1,r , with C independent of ε or f. ( . )
In view of Step , in order to obtain ( . ) it su fices to establish, with C = C(N, j, r), the estimate

ˆQε ˆRN |∇f t,ε (x)| r dxdt ĺ Cε N ˆRN |∇f (x)| r dx. ( . )
We next observe that it su fices to prove ( . ) when f ∈ C ∞ c . Indeed, assuming for the moment that ( . ) holds for such f , Step combined with ( . ) for f ∈ C ∞ c and with a standard limiting argument implies that ( . ) holds for every f ∈ W 1,r .

We finally turn to the proof of ( .

) when f ∈ C ∞ c . Let, for t ∈ R N and ε > 0, v := t+(ε, . . . , ε) and C := C N -j-1,v,ε . Then (see Lemma . ) the projection R N \ C t + 2εk + x N Ψ Þ -→ t + 2εk + x j ∈ C j,t,ε , ∀ k ∈ Z N , ∀ x N ∈ Q ε
is well-defined, locally Lipschitz, and satisfies

|∇Ψ(x)|À ε d(x, C ) . ( . )
It follows from ( . ) and the fact that C is locally a finite union of (N -j -1)-planes that

|∇Ψ|∈ L r loc (R N ), ∀ 1 ĺ r < j + 1. ( . )
Combining ( . ), the fact that f ∈ C ∞ c , Lemma . , and the observation that

f t,ε = f • Ψ, we find that f t,ε ∈ W 1,r (R N ), ∀ 1 ĺ r < j + 1, and df t,ε = [(df ) • Ψ] dΨ in the sense of distributions.
A ter these preliminary remarks, we proceed to the proof of ( . ). By symmetries of the formula defining Ψ, it su fices to establish ( . ) when R N is replaced by

R N * := ∪ k∈Z N (t + 2εk + Q * ε ), with Q * ε := {x N ∈ Q ε ; x 1 ľ • • • ľ x N -j ľ |x m |, ∀ m > N -j}.
We note that, when

x N ∈ Q * ε \ C , we have, ∀ k ∈ Z N , Ψ(t + 2εk + x N ) =t + 2εk + z(x N ), with z(x N ) := (ε, . . . , ε, εx N -j+1 /x N -j , . . . , εx N /x N -j ), ( . )
and

|∇Ψ(t + 2εk + x N )|À ε x N -j , ∀ x N ∈ Q * ε . ( . )
Using ( . ) and ( . ), we find that

ˆQε ˆRN * |∇f t,ε (x)| r dxdt = ˆQε k∈Z N ˆt+2εk+Q * ε |∇f t,ε (x)| r dxdt À ˆQε k∈Z N ˆt+2εk+Q * ε ε r x r N -j |∇f (t + 2εk + z(x)| r dxdt =||∇f || r r ˆQ * ε ε r x r N -j dr ∼ ε N ||∇f || r r ,
where the last line uses the definition of Q * ε and the fact that r < j + 1. We find that ( . ) holds for R N * and thus, as explained above, for R N .

Step is now completed.

Step . Average estimate for f -f t,ε and conclusion. (Here, we use 1 < p < ∞ and j ľ 1.) Let 0 < s < 1, 1 < p < ∞, and 1 ĺ j ĺ N -1 be such that sp < j + 1. We claim that there exist q, r such that

1 < q < ∞, 1 < r < j + 1, 1 p = s r + 1 -s q . ( . )
Indeed, the existence of r and q as in ( .) is equivalent to

s j + 1 + 1 -s ∞ < 1 p < s 1 + 1 -s 1 ,
which clearly holds.

We next recall three classical interpolation results. Given two Banach spaces X and Y , we use the standard notation [X, Y ] s,p ; see e. g. [ , Section . ]. First, when ( . ) holds we have [ , Section . . , Theorem (a), eq. ( ), p. ]

[W 1,r , L q ] s,p = W s,p . ( . )
Next, if X and Y are Banach spaces and s, p, q, r are as above, then [ , Section . . , Theorem, eq. ( ), p.

]

[L r (Ω; X), L q (Ω; Y )] s,p = L p (Ω; [X, Y ] s,p ). ( . )
By ( . ) and ( . ),

∀ r, q as in ( . ), [L r (Q ε ; W 1,r (R N )), L q (Q ε ; L q (R N ))] s,p = L p (Q ε ; W s,p (R N )). ( . )
Final classical result. Let s, p, q, r, X, and Y be as above. Let F be a linear continuous operator from X into L r (Ω; X) and from

Y into L q (Ω; Y ). Then F is linear continuous from [X, Y ] s,p into L p (Ω; [X, Y ] s,p
) and satisfies the norm inequality

F L ([X,Y ]s,p;L p (Ω;[X,Y ]s,p)) ĺ F s L (X;L r (Ω;X)) F 1-s L (Y ;L q (Ω;Y )) . ( . )
By ( . ), ( . ), and ( . ), we find that

F ε (f ) L p (Qε;W s,p (R N )) ĺ C ε N/p f W s,p (R N ) , with C independent of ε. ( . )
(In principle, the constant C in ( . ) may depend on ε, since we apply the interpolation result ( . ) in an ε-dependent domain. The fact that C does not depend on ε is obtained by a straightforward scaling argument: we consider, instead of F ε , the map

G ε (f ) : Q 1 × R N → R , G ε (f )(t, x) = f εt,ε (x).
We obtain ( . ) by applying ( . ) to

G ε (f ) in Q 1 .) A clear consequence of ( . ) is 1 ε N ˆQε ||f t,ε -f || p W s,p (R N ) dt ĺ C||f || p W s,p (R N ) . ( . )
Arguing as above and using ( . ) instead of ( . ), we improve ( . ) to

lim ε→0 1 ε N ˆQε ||f t,ε -f || p W s,p (R N ) dt = 0. ( . )
Clearly, ( . ) and a mean-value argument yield the conclusion of Theorem . .

QED

. . Smoothing in W 1,p

We start by explaining how Theorem . is used in the proof of Theorem . (when

0 < s < 1). Let 0 < s < 1, 1 < p < ∞, 1 ĺ j ĺ N -1 be such that sp < j + 1. (Here and in what follows, j is fixed.) Let f ∈ W s,p (R N ; R ).
It will be convenient here to consider f as a everywhere defined Borel map (rather than an equivalence class). By ( . ), for a.e. t ∈ Q ε we have

f t,ε,j ∈ W s,p . ( . )
On the other hand (by a "generalized slicing" argument, see e. g. [ , Lemma . ]) for a.e. t ∈ Q ε we have

f |Cm,t,ε ∈ W s,p (C m,t,ε ), ∀ 0 ĺ m ĺ N -1.
( . )

(The discussion here being rather informal, we do not give the precise definition of the space W s,p on a skeleton. We will be precise in the case s = 1 detailed below.) Moreover, when sp > 1, we have, for a.e.

t ∈ Q ε [ , Appendix E], tr(f |Cm,t,ε ) = f |C m-1,t,ε , ∀ 1 ĺ m ĺ N. ( . )
Note the assumption sp > 1, which implies that trace theory makes sense in W s,p . (The assumption sp > 1 can be relaxed to sp ľ 1, provided we replace, when sp = 1, the notion of trace with the one of good restriction; see [ , Appendix B, Appendix E].) Combing these facts with ( . ), we find that there exists t = t ε ∈ Q ε such that ( . )-( . ) hold and, in addition f t,ε,j → f in W s,p as ε → 0.

Assume now that we start from u ∈ W s,p (Ω; N ), that we first extend by reflexions to (-2, 3) N , next to a map f ∈ W s,p (R N ; R ). Then, by construction, for small ε,

f t,ε,j is N -valued in (-1, 2) N .
( . )

Up to now, the fact that N is a manifold was irrelevant. The next step consists of taking advantage of the smoothness of N and of the properties ( . )-( . ). More specifically, if: (i) 1 ĺ j ĺ sp < j +1; (ii) ε is fixed; (iii) f ∈ W s,p and t are such that ( . )-( . ) hold, then one may prove that it is possible to approximate, in W s,p ((-1/2, 3/2) N ), f t,ε,j with a map v:

(j) N -valued; (jj) locally Lipschitz in [-1/2, 3/2] \ C N -j-1,v,ε ; (jjj) satisfying |∇v(x)|ĺ C(ε)/d(x, C N -j-1,v,ε ).
By Lemma . , we have v ∈ W 1,q , ∀ q < j + 1. Granted the existence of v as above, we thus obtain that: under the assumptions 0 < s < 1, 1 ĺ sp < N , and 1 ĺ q < sp + 1, each u ∈ W s,p ((0, 1) N ; N ) can be approximated in W s,p with maps v ∈ W 1,q ((0, 1) N ; N ) such that each v is locally Lipschitz outside a finite union of (N -sp -1)-planes. Then a rather standard smoothing procedure (see, e. g., Brezis and Li [ , Proposition A. ] when s = 1) allows to further smoothen the v's and obtain approximation with maps in the class R.

To summarize, the heart of the transition from f t,ε,j to maps in the class R is the construction of v as above. When 0 < s < 1, this is performed in [ , Section ], following a scheme conceptually similar to the one of Hang and Lin for s = 1 [ , Section . , Section ]. In order to keep the presentation technically simple but yet relevant concerning the main ideas, we present here the s = 1 counterpart of the above, consistent with the schemes in [ , ]. More specifically, we will prove the following.

Proposition . . Let ε = 1/2 and t = 0. Let M ∈ N and Ω = (-M -1/2, M + 1/2) N . Let 1 ĺ j < p < j + 1 ĺ N . Let u ∈ W 1,p (Ω; N ) be such that u |C m,0,1/2 ∈ W 1,p (C m,0,1/2 ), ∀ 1 ĺ m ĺ j, ( . ) tr(u |C m,0,1/2 ) = u |C m-1,0,1/2 , ∀ 1 ĺ m ĺ j. ( . )
Then, for every λ > 0, there exists some Lipschitz map g : C j,0,1/2 → N such that the jhomogeneous extension v = g j of g (given by ( . )) satisfies u 0,1/2,j -v W 1,p < λ. † Before proceeding to the proof of Proposition . , let us precise some notation and assumptions. With an abuse of notation, C j = C j ∩C , 0 ĺ j ĺ N -1, where C := ∪ k∈Z N ,|k|ĺM Q 1/2 (k) is the part of the grid C N = C N,0,1/2 corresponding to Ω. C denotes a generic cube in C m . The meaning of ( . )-( . ) is that, for each 1 ĺ m ĺ j, and each cube C of C m , u |C belongs to W 1,p (C) and, in addition, the trace of u |C to ∂C is u |∂C . (Recall that we consider everywhere defined maps.) We naturally define

||u|| p L p (Cm) := C∈Cm ||u|| p L p (C) , |u| p W 1,p (Cm) = ||∇u|| p L p (Cm) := C∈Cm ||∇u|| p L p (C) , ||u|| W 1,p (Cm) := ||u|| L p (Cm) + |u| W 1,p (Cm) .
In view of Lemma . , the conclusion of Proposition . follows from the following fact, that we will establish below: if 1 ĺ j < N , j < p < ∞, and g :

C j → N satisfies g |C m,0,1/2 ∈ W 1,p (C m,0,1/2 ), ∀ 1 ĺ m ĺ j, ( . ) tr(g |C m,0,1/2 ) = g |C m-1,0,1/2 , ∀ 1 ĺ m ĺ j, ( . ) then ∀ λ > 0, ∃ r g ∈ Lip(C j ; N ) s. t. ||r g -g|| W 1,p (C j ) < λ. ( . )
(Note the wider range j < p < ∞ instead of j < p < j + 1.) † Proposition . still holds when p = j, but when j ľ 2 the case p = j requires a separate argument (in the spirit of the proof of Proposition . ), since the embedding W 1,p (R j ) → C 0 holds for p > j, but fails for p = j. For simplicity, we do not consider here the case p = j.

Proof of Proposition . . Step . Choice of a continuous representative. Assume that 1 ĺ j ĺ N and p > j. Assume that g satisfies ( . )-( . ). We claim that there exists a continuous function r g : C j → N such that r g |Cm = g |Cm H m -a. e., ∀ 0 ĺ j ĺ m. The construction of r g is performed successively on each C m , by induction on m. For m = 0, we simply let r g = g. Assuming r g constructed on C m-1 , with 1 ĺ m ĺ j, we let, for C ∈ C m , r g be the continuous representative of g on C. It is easy to see that r g |C m-1 agrees with the map already constructed on C m-1 , is continuous, and that the final map constructed on C j has all the required properties. From now on, we assume that g is continuous.

Step . Reduction to almost N -valued maps. Fix some small δ > 0 such that the projection Π : N δ → N is well-defined and smooth in the δ-neighborhood N δ of N . Assume that we are able to construct a Lipschitz map r g :

C j → N δ such that ||r g -g|| W 1,p (C j ) < λ. Then, clearly, Π • r
g is Lipschitz and, by Lemma . , ||(Π • r g) j -g|| W 1,p < F (λ), for some function F such that lim λ→0 F (λ) = 0. In conclusion, it su fices to prove ( . ) in the apparently weaker form

∀ λ > 0, ∃ r g ∈ Lip(C j ; N δ ) s. t. ||r g -g|| W 1,p (C j ) < λ. ( . )
Step . Approximation on a fixed cube.

Let g ∈ W 1,p (C j ; R ). Let C be a cube in C j , of center 0 C . The projection of the point 0 C + x j ∈ C on ∂C is 0 C + x j-1 , where x j-1 = x j /(2|x j |).
We first define convenient approximations of g as follows. For 0 < µ < 1, we set, with the above notation,

g µ (0 C + x j ) =    g(0 C + x j-1 ), if |x j |ľ (1 -µ)/2 g ˆ0C + x j 1 -µ ˙, if |x j |< (1 -µ)/2 .
Note that g µ is continuous on C j , N -valued, and clearly satisfies ( . )-( . ). The following fact is straightforward.

[1 ĺ p < ∞, 1 ĺ j ĺ N, C ∈ C j , g ∈ W 1,p (C)] =⇒ g µ → g in W 1,p (C) as µ → 0. ( . ) Let ρ ∈ C ∞ c ((-1/2, 1/2) j ) be a standard mollifier. Given h ∈ L 1 (C j ; R ), the convolution h * ρ t is well-defined and smooth in the set ∪ C∈C j {0 C + x j ; |x j |ĺ (1 -t)/2}. (Here, we naturally identify each C with a subset of R j .) Fix some function η ∈ C ∞ c ([0, 1/2); [0, 1]
) such that η(θ) = 1 for small θ. For small t, the map

C 0 C + x j → g t (0 C + x j ) := η(|x j |) g * ρ t (0 C + x j )
is well-defined and smooth in C.

We also set

g 0 (0 C + x j ) := η(|x j |) g(0 C + x j ).
The following is straightforward.

[1 ĺ p < ∞, 1 ĺ j ĺ N, C ∈ C j , g ∈ W 1,p (C)] =⇒ g t → g 0 in W 1,p (C) as t → 0. ( . )
Given f : ∂C → R , set (with H j the homogeneous extension from C j-1 to C j )

T (f )(0 C + x j ) := (1 -η(|x j |)) H j (f )(0 C + x j ), ∀ 0 C + x j ) ∈ C. ( . )
We note the following consequence of (the proof of) Lemma . .

[1 ĺ p < ∞, 1 ĺ j ĺ N, C ∈ C j ] =⇒ [W 1,p (∂C) f → T (f ) ∈ W 1,p (C) is continuous]. ( . )
Finally, let us note the following consequence of the embedding W 1,p (C) → C 0 , valid when p > j.

[1 ĺ j < p < ∞, 1 ĺ j ĺ N, C ∈ C j , g ∈ W 1,p (C; N )] =⇒ ∃ t 0 s. t. [t ĺ t 0 , |x j |ĺ (1 -t)/2 =⇒ g * ρ t (0 C + x j ) ∈ N δ ]. ( . )
(The validity of ( . ) when p = j requires a separate argument, relying on Lemma . .)

Step . Proof of ( . ) when j = 1. By ( . ), it su fices to prove ( . ) when g is replaced by g µ . Since j = 1 and thus C 0 is a finite collection of points, we may thus assume that g is constant near each point in C 0 :

∃ 0 < µ < 1 s. t. [C ∈ C 1 , |x 1 |ľ (1 -µ)/2] =⇒ g(0 C + x 1 ) = g(0 C + x 0 )]. ( . ) Let now η ∈ C ∞ ([0, 1/2); [0, 1]) be such that η(θ) = 1, if 0 ĺ θ ĺ 1/2 -µ/4 0, if θ > 1/2 -µ/6 . ( . )
When 0 < t ĺ µ/6, the map

C 1 x = 0 C + x 1 → G t (x) = η(|x 1 |) g * ρ t (0 C + x 1 ) + `1 -η(|x 1 |) ˘g(0 C + x 0 )
is well-defined everywhere on C 1 , and is Lipschitz. Moreover, by ( . ) and the choice of η, we clearly have G t → g in W 1,p as t → 0.

It remains to prove that, for small t, we have

G t (0 C + x 1 ) ∈ N δ , ∀ C ∈ C 1 , ∀ 0 C + x 1 ∈ C. ( . )
By ( . ) and ( . ), property ( . ) holds when |x 1 |ĺ 1/2-µ/5. Clearly, ( . ) holds also when |x 1 |ľ 1/2 -µ/6. Finally, when 1/2 -µ/5 ĺ |x 1 |ĺ 1/2 -µ/6 and t ĺ µ/6, we have

G t (0 C + x 1 ) = g * ρ t (0 C + x 1 ) = g(0 C + x 0 ) ∈ N .

This completes Step .

Step . Proof of Proposition . by induction on j. (Here, we use the assumptions ( . )-( . ) at all dimensions 1 ĺ m ĺ j.) Let 2 ĺ j ĺ N . Let f be the restriction of g to C j-1 . By ( . ), we may assume that there exists some µ ∈ (0, 1) such that

g(0 C + x j ) = f (0 C + x j-1 ), ∀ C ∈ C j , ∀ 0 C + x j ∈ C s. t. |x j |ľ (1 -µ)/2.
( . ) By ( . )-( . ) and the induction hypothesis, the map f is the limit in W 1,p of a sequence (F k ) ⊂ Lip(C j-1 ; N ). With η as in ( . ) and 0 < t ĺ µ/6, we define, everywhere on C j , the Lipschitz maps

C j 0 C + x j → G k,t (0 C + x j ) :=η(|x j |) g * ρ t (0 C + x j ) + (1 -η(|x j |)) F k (0 C + x j-1 ).
By ( . ) and ( . ), we have

lim k→∞ lim t 0 G k,t = g in W 1,p (C j ).
In order to complete Step and the proof of Proposition . , it remains to prove that, for large k and su ficiently small t (possibly depending on k) we have

G k,t (0 C + x j ) ∈ N δ , ∀ C ∈ C j , ∀ 0 C + x j ∈ C. ( . )
As in Step , ( . ) holds when |x j |ĺ 1/2 -µ/5 or |x j |ľ 1/2 -µ/6. When 1/2 -µ/5 ĺ |x j |ĺ 1/2 -µ/6, we argue as follows. By the Sobolev embeddings, we have

F k → f uniformly. Let k 0 be such that F k -f ∞ ĺ δ, ∀ k ľ k 0 .
( . ) By ( . ) and the continuity of f , for every fixed k we have

lim t 0 G k,t (0 C +x j ) = η(|x j |) f (0 C + x j-1 ) + (1 -η(|x j |)) F k (0 C + x j-1 )
uniformly in the set

C∈C j {0 C + x j ; 1/2 -µ/5 ĺ |x j |ĺ 1/2µ/6}. ( . )
We complete the proof of ( . ) using ( . ) and ( . ).

QED

. . Singularities removing technique in W 1,p

One of our purposes here is the proof of Theorem . when 0 < s < 1, under the necessary condition that π sp (N ) is trivial. We have seen in Sections . and . that maps of the form g j , where g ∈ Lip(C j ; N ), are dense in W s,p (Ω; N ), at least when 1 ĺ j < sp < j + 1 ĺ N .

We have already noted that g j actually belongs to the space W 1,q (Ω; N ), ∀ 1 ĺ q < j + 1. We will prove below that g j can be approximated, in W 1,q (Ω), ∀ 1 ĺ q < j + 1, with Lipschitz Nvalued maps. This fact, combined with the Gagliardo-Nirenberg inequalities (Corollary . ) and a straightforward smoothing argument, implies Theorem . when 0 < s < 1.

A ter these introductory remarks, we present and prove the main result of this section (see Bethuel [ ], with roots in White [ , Section ], for the main idea of the proof (Step below) and, for the presentation we give here, also Hang and Lin [ , Section ] and Bousquet, Ponce, and Van Scha tingen [ , Section ]). The result is stated, with no loss of generality, in Ω = (-M -1/2, M + 1/2) N . Proposition . . Let 1 ĺ j ĺ N -1 and 1 ĺ q < j + 1. Assume that π j (N ) is trivial. Then, for every g ∈ Lip(C j ; N ), the map g j is strong limit in W 1,q of maps in Lip(Ω; N ).

Proof.

Step . Construction of a Lipschitz N -valued extension h of g to C j+1 . (Here, we use the assumption on π j (N ).) Let C ∈ C j+1 . Since ∂C is bi-Lipschitz homeomorphic with S j , and, by assumption, π j (N ) is trivial, there exists a homotopy

G C : ∂C × [0, 1] → N such that G C (x, 0) = g(x), ∀ x ∈ ∂C, and G C (x, θ) = b C , ∀ x ∈ ∂C, ∀ θ ľ 1/2, for some constant b C ∈ N .
Moreover, by a smoothing argument, we may assume that G C is Lipschitz. The map

C j+1 h(0 C + x j+1 ) := G C (0 C + x j , 1 -2|x j+1 |), if |x j+1 |ľ 1/4 b C , if |x j+1 ĺ 1/4 is a Lipschitz N -valued extension of g to C j+1 .
Step . Construction of a Lipschitz N -valued extension k of g to C N . (Here, we use the existence of the map h from the previous step.) We rely on the following geometrically obvious fact (see Lemma . for a formal proof). There exists a Lipschitz homotopy G = G(x, θ) :

C N × [0, 1] → C N such that: a) G(x, 0) = x, ∀ x ∈ C N . b) G(x, θ) = a, for some (fixed) point a ∈ C j+1 , ∀ x ∈ C N , ∀ θ ľ 1/2. c) G(x, θ) ∈ C j+1 , ∀ x ∈ C j , ∀ θ.
Granted the existence of G, and with h as in Step , we let, ∀ 0

C + x N ∈ C ∈ C N , k(0 C + x N ) := h(G(0 C + x j , 2d(0 C + x N , C j ))), if d(0 C + x N , C j ) ĺ 1/4 h(a), if d(0 C + x N , C j ) ľ 1/4 .
Clearly, k is a Lipschitz N -valued extension of g to C N .

Step . Approximation of g j . (Here, we use the assumption q < j + 1.) For 0 < µ < 1/2, consider the following sets and functions:

U µ := {x ∈ C N ; d(x, C j ) ĺ 1/2 -µ}, V µ := {x ∈ C N ; 1/2 -µ ĺ d(x, C j ) ĺ 1/2 -µ/2}, W µ := {x ∈ C N ; d(x, C j ) ľ 1/2 -µ/2}, f 1 : [1/2 -µ, 1/2] → [0, 1], f 1 (θ) := 0, if θ ľ 1/2 -µ/2 (1 -2θ)/µ -1, if 1/2 -µ ĺ θ ĺ 1/2 -µ/2 , f 2 : [1/2 -µ, 1/2] → [0, 1], f 2 (θ) := 1, if θ ľ 1/2 -µ/2 -(1 -2θ)/µ + 2, if 1/2 -µ ĺ θ ĺ 1/2 -µ/2 , d j (x) := d(x, C j ).
We define the following approximation of g j :

C N x = 0 C + x N → F µ (x) := g j (x) = g(0 C + x j ), if x ∈ U µ , k(f 1 (d j (x))(0 C + x j ) + f 2 (d j (x))x), if x ∈ V µ ∪ W µ .
We note that F µ is well-defined, Lipschitz, N -valued, equals g j in U µ , and is Lipschitz (with Lipschitz constant independent of µ) in W µ . Since |W µ |→ 0 as µ → 0, in order to prove that F µ → g j in W 1,q (Ω) as µ → 0 it remains to prove that ||∇F µ || L q (Vµ) → 0 as µ → 0. In turn, this follows from the fact that, from the definition of F µ and the fact that k is Lipschitz, we have ˆVµ

|∇F µ | q À 1 µ q |V µ |∼ µ j+1 µ q → 0 as ε → 0. QED Lecture # .

Hearing singularities

Let us return to the all purposes counterexample in Proposition . . It relies on the existence of a non-trivial topological invariant (in that case, the winding number of maps f ∈ C 0 (S 1 ; S 1 )) and on the construction of a map "carrying" the topological invariant around a singular point. This raises several questions: In this section, we discuss the best understood situation, the one of sphere-valued maps. In order to further simplify the presentation and focus on analytical (rather than geometrical measure theory) issues, we first assume that the space dimension N and the dimension k of the sphere are related by N = k +1. In Section . , we provide a glimpse of the general case and of the additional di ficulties it raises.

Contents

. . The distributional Jacobian

We let, in Sections . -. , u : Ω → S N -1 , where Ω = (0, 1) N and N ľ 2. Recall that we always assume that sp < N . If sp < N -1, then C ∞ (Ω; S N -1 ) is dense in W s,p (Ω; S N -1 ) (by Theorem

. and the fact that π j (S N -1 ) is trivial when j < N -1). Therefore, the interesting range is

N -1 ĺ sp < N. ( . )
For such s and p, maps in the class R = R 0 (i. e., maps as in ( . ), with A a finite subset of Ω) are dense in W s,p (Ω; S N -1 ) (Theorem . ). When u ∈ R, one can define the singular set simply as A. However, this is not a tractable definition, since it is not clear how to pass to the limits sets of points. The appropriate substitute is the distribution

Ju := C N a∈A deg(u, a)δ a ∈ D (Ω), ( . )
where

C N := |B 1 (0)| (the volume of the unit ball in R N ).
Here, deg(u, a) is the (Brouwer) degree of the map u |Sε(a) : S ε (a) → S N -1 , for small ε. Clearly, this integer does not depend on (small) ε.

The main result here is the following (see [ ] for the full result, and, for special cases, Bethuel, Brezis, and Coron [ ], Jerrard and Soner [ , ], Hang and Lin [ ]).

Theorem . . Assume ( . ). Then the map

R u J Þ -→ Ju ∈ D (Ω) ( . )
has a unique extension by continuity, still denoted J, to W s,p (Ω; S N -1 ).

In addition, Ju belongs to the space [Lip 0 (Ω)] * , the mapping W s,p (Ω; S N -1 ) u → Ju ∈ [Lip 0 (Ω)] * is continuous, and we have the estimate

||Ju|| [Lip 0 (Ω)] * À |u| (N -1)/s W s,p .
( . )

Proof.

Step . A first convenient formula for Ju. We first derive a tractable formula for Ju when u ∈ R. This formula (which will explain the title of this section) appears in Brezis, Coron, and Lieb [ ], with roots in Ball [ ] and Morrey [ ]. To start with, we note that R ⊂ W 1,q , ∀ q < N . (For this step, R ⊂ W 1,N -1 su fices.) Let ω = ω N -1 be the standard volume form on S N -1 , given by

ω N -1 := N j=1 (-1) j-1 x j dx 1 ∧ . . . ∧ dx j ∧ . . . ∧ dx N . ( . )
Denoting u ω the pullback by u of ω, i. e.,

u ω = N j=1 (-1) j-1 u j du 1 ∧ . . . ∧ du j ∧ . . . ∧ du N ∈ L 1 (Ω; Λ N -1 ),
we claim that

Ju = 1 N d(u ω) in D (Ω), ( . )
where, in ( . ), we have identified a scalar distribution (the le t-hand side) with a N -form whose density is a distribution (the right-hand side).

To justify ( . ), a first important fact is that, when u is C 2 in some open set V ⊂ Ω, we have, in V ,

d(u ω) = N du 1 ∧ • • • ∧ du N = N (Jac u) dx 1 ∧ • • • ∧ dx N = 0, ( . )
where Jac stands for the Jacobian determinant. The first equality is a clear consequence of the exterior calculus rules, and justifies the designation of d(u ω) as (up to a constant factor N ) distributional Jacobian. The last equality is justified by the fact that the N vectors d 1 u(x), . . . d N u(x),

x ∈ V , are all in the (N -1)-dimensional tangent hyperplane T S N -1 (x), and thus Jac u(x) = 0, ∀ x ∈ V .

A second important fact is Kronecker's formula (see, e. g., Dinca and Mawhin [ , Section . , Section . ]): if S is any sphere in R N (with the usual orientation), then For further use, let us note that we have proved that, when u ∈ R, we have

deg(v, S) = 1 |S N -1 | ˆS v ω, ∀ v ∈ C 1 (S; S N -1 ). ( . ) Let now u ∈ R. Combining: (i) the definition of the distributional derivative; (ii) the fact that u ∈ W 1,N -1 ; (iii) ( . ); (iv) the divergence theorem; (v) the fact that u ∈ R; (vi) ( . ), we find that, ∀ ϕ ∈ C ∞ c (Ω), d(u ω)(ϕ) = -ˆΩ dϕ ∧ (u ω) = -lim ε→0 ˆΩ\∪ a∈A Bε(a) dϕ ∧ (u ω) = -lim ε→0 ˆΩ\∪ a∈A Bε(a) d[ϕ(u ω)] =
Ju(ϕ) = - 1 N ˆΩ dϕ ∧ (u ω), ∀ ϕ ∈ C ∞ c (Ω), ( . ) and that, if u ∈ C 2 (Ω; R N ), we have ˆΩ(Jac u)ϕ = - 1 N ˆΩ dϕ ∧ (u ω), ∀ ϕ ∈ C ∞ c (Ω; R). ( . )
Step . The easy case s ľ 1. The right-hand side of ( . ) is clearly continuous (with respect to the u's satisfying |u|ĺ 1) in W 1,N -1 . We complete this step by noting that, when s ľ 1 and sp ľ N -1, we have W s,p ∩ L ∞ → W 1,N -1 (Corollary . ).

In the remaining part of the proof, we assume that 0 < s < 1.

Step . A second convenient formula for Ju. This step appears in [ ], but was essentially known before (see Dunford and Schwartz [ , p. ]).

Let u ∈ C 2 (Ω; R N ), respectively ϕ ∈ C ∞ c (Ω; R). Let W = W (u) ∈ C 2 (Ω × [0, 1); R N ) be an extension of u, respectively Φ = Φ(ϕ) ∈ C 1 c (Ω × [0, 1
); R) be an extension of ϕ. Let, for each j ∈ {1, . . . , N + 1}, E j =E j (W ) denote the determinant whose columns are the N partial derivatives ∂ 1 W, . . . , z ∂ j W ,. . . , ∂ N +1 W . We claim that

N +1 j=1 (-1) j-1 ∂ j E j = 0. ( . ) Indeed, identifying (E 1 , . . . , E N +1 ) with the N -form ζ := dW 1 ∧ • • • ∧ dW N , ( .
) amounts to the trivial equality dζ = 0 (see [ , Lemma . ] for details).

Combining: (i) ( . ); (ii) the divergence theorem; (iii) the fact that, on Ω × {0}, we have E N +1 Φ = (Jac u)ϕ; (iv) ( . ), we find that ˆΩ×(0,1)

N +1 j=1 (-1) N +j E j ∂ j Φ = ˆΩ×(0,1) N +1 j=1 ∂ j ((-1) N +j E j Φ) = ˆΩ×{0} E N +1 Φ = ˆΩ(Jac u)ϕ = - 1 N ˆΩ dϕ ∧ (u ω), so that 1 N ˆΩ dϕ ∧ (u ω) = - ˆΩ×(0,1) N +1 j=1 (-1) N +j E j ∂ j Φ. ( . )
At this stage, we know that ( . ) holds when u ∈ C 2 and W ∈ C 2 . By a straightforward argument, ( . ) still holds provided ϕ ∈ Lip 0 (Ω), Φ ∈ Lip 0 (Ω × [0, 1)) and

u ∈ W 1,N -1 loc (Ω; R N ) ∩ L ∞ , W ∈ W 1,N loc (Ω × (0, 1); R N ) ∩ L ∞ , W (•, ε) → u in W 1,N -1 loc (Ω) as ε → 0. ( . )
Combining the first two steps, we find the useful identity

Ju(ϕ) = - ˆΩ×(0,1) N +1 j=1 (-1) N +j E j ∂ j Φ, ∀ u ∈ R, ∀ W as in ( . ), ( . )
and also the fact that the right-hand side of ( . ) is well-defined (under the assumption ( . )) for Φ ∈ Lip 0 (Ω × [0, 1)).

The heart of the proof of Theorem . consists of proving the existence, for each u ∈ R, of a convenient extension W = W (u) such that the right-hand side of ( . ) is continuous in W s,p (with respect to u).

Step . The main geometric estimate. (Here, we do not use the assumption sp ľ N -1.) Consider a linear continuous operator W s,p u = u(x) Section . ). By slicing, trace theory, and Sobolev embeddings, for a.e. x ∈ Ω we have (j)

→ U = U (x, ε), x ∈ Ω, 0 < ε ĺ 1 such that: (i) U ∈ C ∞ ; (ii) U ∈ W s+1/p,p ; (iii) tr U = u; (iv) |U | W s+1/p,p À |u| W s,p ; (v) ||U || ∞ À ||u|| ∞ if u ∈ L ∞ ; (vi) if u ∈ W σ,q , then U (•, ε) ∈ W σ,q and |U (•, ε)| W σ,q À |u| W σ,q (see
lim ε→0 U (x, ε) = u(x) ∈ S N -1 ; (jj) U (x, •) ∈ W s+1/p,p ((0, 1)) → C s ([0, 1]). Define the function d(x) := inf{0 < ε ĺ 1; |U (x, ε)|ĺ 1/2}, with the convention that inf ∅ = ∞. We claim that ˆΩ 1 [d(x)] sp dx À |u| p W s,p . ( . )
Indeed, if x is such (j) and (jj) hold and d(x) < ∞, then d(x) > 0 and |U (x, d(x))|= 1/2, and therefore

1/2 ĺ |u(x) -U (x, d(x))|À [d(x)] s |U (x, •)| C s ([0,1]) À [d(x)] s |U (x, •)| W s+1/p,p ((0,1)) . ( . )
Using ( . ), slicing, and (iv), we find that

ˆΩ 1 [d(x)] sp dx À ˆΩ|U (x, •)| p W s+1/p,p ((0,1)) dx À |u| p W s,p ,
so that ( . ) holds, as claimed.

Step . Construction of W (u) and Φ(ϕ). Fix some

ζ ∈ C 1 c ([0, 1); R) such that ζ(0) = 1. Let Φ(ϕ)(x, ε) := ζ(ε)ϕ(x), ∀ x ∈ Ω, ∀ ε ∈ [0, 1]. Clearly, ϕ → Φ is linear and continuous from Lip 0 (Ω) into Lip 0 (Ω × [0, 1)).
Let Π ∈ C ∞ (R N ; R N ) be such that Π(x) = x/|x| when |x|ľ 1/2. (Here, | | stands for the Euclidean norm.) Let U be as in the previous step. Set W := Π • U . If u ∈ R, then u ∈ W 1,N -1 and, by the construction of U , we have ||∇U (•, ε)|| N -1 À ||∇u|| N -1 . By the formula of W , we also have ||∇W (•, ε)|| N -1 À ||∇u|| N -1 . On the other hand, we have U (•, ε) → u in W 1,N -1 as ε → 0 and U ∈ L ∞ . By Theorem . and the fact that Π • u = u, we find that W (•, ε) → u in W 1,N -1 as ε → 0. Therefore, ( . ) holds, and thus ( . ) holds when u ∈ R and W = W (u).

Step . Conclusion. It remains to prove that the right-hand side of ( . ) is continuous from R (with the distance inherited from W s,p ) into [Lip 0 (Ω)] * . Consider the open set

V = V (u) :={(x, ε) ∈ Ω × (0, 1); |U (x, ε)|> 1/2} ⊂{(x, ε) ∈ Ω × (0, 1); 0 < ε < min{d(x), 1}}.
In V , we have |W |= 1, and thus E j = 0, ∀ j (see the proof of ( . )). On the other hand, for (x, ε) ∈ V , we have, by the construction of U (see Section . )

|∇W (x, ε)|À |∇U (x, ε)|À 1 ε ||u|| ∞ À 1 ε . ( . )
Using: (i) ( . ); (ii) ( . ); (iii) the fact that, when (x, ε) ∈ V , we have ε ľ d(x); (iv) ( . ), we find that

|Ju(ϕ)|À||∇Φ|| ∞ ˆΩ ˆεľd(x) 1 ε N dεdx À ||∇ϕ|| ∞ ˆΩ 1 [d(x)] N -1 dx À||∇ϕ|| ∞ |u| (N -1)/s W s,p , ( . )
where the last line uses ( . ), Hölder's inequality, and the assumption sp ľ N -1.

In order to complete the proof, it su fices to prove the continuity of R u → Ju(ϕ) for a fixed ϕ. Consider a sequence (u j ) ⊂ R converging in W s,p to some u. By: (i) trace theory; (ii) slicing; (iii) the converse to the dominated convergence theorem, there exist a subsequence, still denoted (u j ), and a function F ∈ L p (Ω) such that |U j (x, •)| W s+1/p,p ĺ F (x) for each j and a. e. x ∈ Ω. An inspection of the proof of ( . ) shows that, for each j and a. e. x, we have d(x) Á [F (x)] -1/s , and therefore the corresponding sets V (u j ) satisfy

[Ω × (0, 1)] \ V (u j ) ⊂ Z := {(x, ε) ∈ Ω × (0, 1); ε Á [F (x)] -1/s }.
( . )

(Note that Z does not depend on the (sub)sequence (u j ).) Using: (i) ( . ); (ii) the fact that, clearly,

W (u j )(x, ε) → W (u)(x, ε), ∀ x, ∀ ε; (iii) ( . ); (iv) the fact that (x, ε) → 1/ε N ∈ L sp/(N -1) (Z) ⊂ L 1 (Z); (v)
dominated convergence, we find that (possibly along a subsequence) (J(u j )(ϕ)) converges to the right-hand side of ( . ) corresponding to u. Finally, the uniqueness of the limit implies that convergence holds for the full original sequence.

Moreover, using the above domination and the explicit construction of Φ(ϕ), the continuity of W s,p (Ω; S N -1 ) u → Ju ∈ [Lip 0 (Ω)] * is routine. The estimate ( . ) easily follows from ( . ) and a limiting argument.

QED

. . The range of the distributional Jacobian

Recall that we consider maps u : Ω ⊂ R N → S N -1 , with N ľ 2. The main result here is the following.

Theorem . . Assume ( . ).

. If u ∈ W s,p (Ω; S N -1 ), then there exist points P j , N j ∈ Ω, j ľ 1, such that

j |P j -N j |À |u| (N -1)/s W s,p , ( .
)

Ju = C N j (δ P j -δ N j ) in D (Ω). ( . )
. Conversely, given points P j , N j ∈ Ω satisfying ( . ) and j |P j -N j |< ∞, there exists u : Ω → S N -1 such that, for every s, p satisfying sp = N -1:

(i) u ∈ W s,p (Ω; S N -1 ); (ii) ( . ) holds; (iii) |u| p W s,p À inf k | r P k -r N k |; k (δ r P k -δ r N k ) = j (δ P j -δ N j ) in D (Ω) . ( . )
See [ , ] for the general case, and [ , , ] for special cases. When s = 1 and p = N -1, the above theorem is a special case of the main result in Alberti, Baldo, and Orlandi [ , Theorem . ], but obtaining Theorem . from [ , Theorem . ] requires an additional argument. The proof involves two important ingredients: a duality formula and a dipole construction, both due to Brezis, Coron, and Lieb [ ], complemented with a dipole insertion technique due to Bethuel [ ].

Remark . . Note the range N -1 ĺ sp < N in item , and the range sp = N -1 in item . Thus, item is not the exact converse of item . When N = 2 (i. e., we consider S 1 -valued maps), the exact converse of item is known (see Bousquet [ ]), i. e., when 1 < sp < 2, it is possible to characterize the set {Ju; u ∈ W s,p (Ω; S 1 )}. The counterpart of the result in [ ] is not known when N > 2.

Elements of proof of Theorem . . Step . A pseudometric and a duality formula. Set, for P, N ∈ Ω,

d(P, N ) = min{|P -N |, dist(P, ∂Ω) + dist(N, ∂Ω)}. ( . )
Clearly, d is a pseudometric, and, for each P, N ∈ Ω: (i) either d(P, N ) = |P -N | and the interior of the segment [P, N ] is completely contained in Ω, or: (ii) there exist points

P 1 , N 1 ∈ ∂Ω such that |P -N 1 |= d(P, ∂Ω), |P 1 -N |= d(N, ∂Ω), and d(P, N ) = |P -N 1 |+|P 1 -N |.
Moreover, in the latter case, if, for example, P ∈ Ω, then the line segment [P, N 1 ] is normal to ∂Ω at N 1 , and its interior is completely contained in Ω.

Given P j , N j ∈ Ω, 1 ĺ j ĺ m, set L((P j ), (N j )) := min σ∈Sm j d(P j , N σ(j) ).

( . )

It is clear from the definitions ( . ) and ( . ) that, given initial collections (P j ), (N j ), 1 ĺ j ĺ m, we may find new collections, still denoted, for simplicity, (P k ), (N k ) (containing, possibly, more points), such that new points

(δ P k -δ N k ) = initial points (δ P j -δ N j ) in D (Ω), ( . ) L((P j ), (N j )) = L((P k ), (N k )) = k |P k -N k |, ( . )
for each k, the points P k , N k are distinct, at least one of them is in Ω, and, if

P k ∈ ∂Ω or N k ∈ ∂Ω, the segment [P k , N k ] is normal to ∂Ω. ( . )
Note that, if ϕ ∈ Lip 0 (Ω) and P, N ∈ Ω, then ( . )

ϕ(P ) -ϕ(N ) ĺ d(P, N )|ϕ| Lip , (
For further use, let us note the following consequence of ( . ) combined with ( . ) below and ( . )-( . ):

||Ju|| [Lip 0 (Ω)] * = C N min k |P k -N k |; P k , N k ∈ Ω,Ju = C N (δ P k -δ N k ) , ∀ u ∈ R 0 .
( . )

Step . Proof of item . We start with a preliminary remark. If u ∈ R then (in view of the definition ( . ) of Ju), possibly a ter adding fictitious points on ∂Ω, we may always write

Ju = C N j (δ P j -δ N j ) in D (Ω) ( . )
(where the sum contains a finite number of terms).

Let now u ∈ W s,p (Ω; S N -1 ). Let u 0 be any constant in S N -1 . Consider a sequence

(u i ) iľ1 ⊂ R such that u i → u in W s,p , |u 1 | W s,p À |u| W s,p , and ||Ju i+1 -Ju i || [Lip 0 (Ω)] * ĺ 2 -i |u| (N -1)/s W s,p
, ∀ i ľ 1 (see Theorem . ). Combining the observation ( . ) with ( . ), ( . )-( . ), and the estimate ( . ), we find that there exist sequences

(P k,i ) k , (N k,i ) k such that Ju i+1 -Ju i = C N k (δ P k,i -δ N k,i ) in D (Ω), ∀ i ľ 0, ( . ) k |P k,0 -N k,0 |À |u| (N -1)/s W s,p , ( . ) k |P k,i -N k,i |À 2 -i |u| (N -1)/s W s,p , ∀ i ľ 1. ( . )
Combining ( . )-( . ) with the continuity of J, we find that ( . )-( . ) hold.

Step . Partial proof of item : setting and strategy. We present the proof of a weaker result: we let N ľ 3 and we fix 0 < s < N -1 and 1 < p < ∞ such that sp = N -1. (For an "all couples" (s, p procedure, based on a diagonal process and Gagliardo-Nirenberg, in a similar context, see [ , proof of Theorem . ].) For such s, p, and N , and all sequences (P j ), (N j ) as in item , we prove the existence of a map u satisfying ( . ). For the proof of item in full generality, we refer the reader to [ ].

For pedagogical purposes, we temporarily assume that

k |P k -N k | 1/p < ∞; ( . )
we will remove this assumption in the final step.

It will be more convenient to work in the full space R N , N ľ 3. More precisely, given sequences

(P k ) kľ1 , (N k ) kľ1 ⊂ R N such that k |P k -N k |< ∞ and P k = N k , ∀ k,
and a point a ∈ S N -1 , we will construct a map u : R N → S N -1 such that:

a) u -a ∈ W s,p (R N ). b) Ju = C N k (δ P k -δ N k ) in D (R N ). c) ||u -a|| p W s,p À k |P k -N k |< ∞.
Clearly, the existence of such a map implies item of the theorem.

The map u will be obtained as the limit of a sequence of maps, each iterative step consisting of dipoles insertions.

Step . The dipole construction. We fix a map

f ∈ C ∞ ([0, 1]; [0, 1]) such that f (0) = f (1) = 0, f (0) > 0, f (1) < 0, and f (θ) > 0, ∀ θ ∈ (0, 1).
Given a line segment S in R N , say, in order to simplify the statement, S = [0, Le N ], and 0 < ε ĺ L, there exists a map

u ε ∈ C ∞ (R N \ {0, Le N }; S N -1 ) such that u ε ∈ R 0 , ∀ ε, ( . ) |u ε | q W σ,q À L, ∀ σ, q such that σq = N -1, ∀ ε, ( .
) Lemmas . and . ). A similar conclusion holds for an arbitrary segment. A noticeable fact is that the estimate ( . ) involves the length of the segment.

Ju ε = C N (δ 0 -δ Le N ) in D (R N ), ∀ ε, ( . ) supp(u ε -a) ⊂ {(x , x N ) ∈ R N -1 × R; 0 ĺ x N ĺ L, |x |ĺ Lεf (x N /L)}, ∀ ε ( . ) (see
Step . The iterative construction. We will construct a sequence (v k ) such that v 0 = a and

Jv m = C N nĺm (δ Pn -δ Nn ) in D (R N ), ∀ m, ( . ) ||v m -v m-1 || p W s,p À |P m -N m |, ∀ m ľ 1, ( . ) ||v m || p W s,p -||v m-1 || p W s,p À |P m -N m |, ∀ m ľ 1. ( . )
Assuming ( . )-( . ) and using the temporary assumption ( . ), we find that the limiting map u := a + mľ1 (v m -v m-1 ) has all the desired properties.

To start with, we let, as in the dipole construction,

v 1 ∈ C ∞ (R N \ {P 1 , N 1 }; S N -1 ) satisfy Jv 1 = C N (δ P 1 -δ N 1
) and the estimates ( . )-( . ). (This is possible, for su ficiently small ε, since P 1 = N 1 .) Assume next that we were able to construct v 1 , . . . , v k-1 such that ( . )-( . ) hold, and, in addition, there exists an increasing sequence of finite sets

A m ⊂ R N , 1 ĺ m ĺ k-1, such that: v m is smooth in R N \ A m , ( . ) v m ∈ R 0 , ( . ) for each x ∈ A m , there exists a non-empty open conical cap C x with vertex x such that v m (x) = a in C x . ( . )
Note that these assumptions are satisfied when m = 1, with A 1 = {P 1 , N 1 }. We next construct v k according to the position of P k and N k with respect to the set A k-1 .

Case . [P

k , N k ] ∩ A k-1 = ∅. We first modify v k-1 in a convenient small open neighborhood V of [P k , N k ],
such that the modified map, still denoted v k-1 , continues to satisfy ( . )-( . ), and v k-1 = a in V . (Intuitively, this is possible since a segment in R N , with N ľ 3, has zero W s,pcapacity.) The rigorous existence of such a modified map is established in Lemma . . We next consider, with an abuse of notation, the map

v k = v k-1 , in R N \ V u ε , in V,
where u ε is the map in the dipole construction corresponding to the singularities P k , N k . Clearly, in view of ( . )-( .) and of the Brezis-Lieb type Lemma . , for small ε the map v k satisfies ( . )-( .), with

A m := A m-1 ∪ {P k , N k }.
Case .

[P k , N k ] ∩ A k-1 = ∅.
In this case, we may construct a finite chain

D = [Q 1 , Q 2 ] ∪ . . . ∪ [Q t-1 , Q t ]
without self intersections and such that:

(i) If x ∈ D \ {P k , N k }, then x ∈ A k-1 ∪ {P k , N k }. In particular, Q 2 , . . . , Q t-1 ∈ A k-1 ∪ {P k , N k }. (ii) If P k ∈ A k-1 , then, near P k , the segment [Q 1 , Q 2 ] is contained in C P k , where C P k is as in ( . ). Similarly for N k . (iii) j |Q j+1 -Q j |À |P k -N k |. We next modify v k-1 in a neighborhood of D \ {P k , N k } such that: (j) ( . )-( . ) still hold; (jj) v k-1 equals a in a neighborhood of D \ {P k , N k }.
The construction of the modified map and the corresponding estimates are established in Lemma . . Finally, we insert (t-1)-dipoles u j,ε j ,

1 ĺ j ĺ t -1, satisfying Ju j,ε j = C N (δ Q j+1 -δ Q j )
. By the multi-sequences Brezis-Lieb lemma . , for convenient small ε j , the new map

v k (x) = a + u j,ε j -a, in supp(u j,ε j -a) v k-1 -a, in R N \ ∪ j supp(u j,ε j -a)
has all the required properties, with

A k = A k-1 ∪ {Q 1 , . . . , Q t }.
Step . Removing the assumption ( . ). Let S :

= k |P k -N k |. We consider integers 1 = j 0 < j 1 < j 2 < . . . such that j k-1 ĺj<j k |P j -N j |À 2 -k S, ∀ k.
We let v 0 = a and construct, as explained in

Step , Case (using several chains and the multi-sequences Brezis-Lieb Lemma . ), a sequence

(v k ) such that Jv k = C N j<j k (δ P j -δ N j ) in D (R N ), ∀ k, ||v k -v k-1 || p W s,p À j k-1 ĺj<j k |P j -N j |À 2 -k S, ∀ k ľ 1, ||v k || p W s,p -||v k-1 || p W s,p À j k-1 ĺj<j k |P j -N j |À 2 -k S, ∀ k ľ 1, v k is smooth in R N \ A k , for some finite A k , v k ∈ R 0 , for each x ∈ A k , there exists a non-empty open conical cap C x with vertex x such that v k (x) = a in C x .
Then (v k -a) converges in W s,p to some map v with v -a ∈ W s,p and such that Jv = C N j (δ P j -δ N j ).

QED

. .

Inserting singularities

Recall that we consider maps u : Ω ⊂ R N → S N -1 , with N ľ 2. The main result here is due to Bethuel when s = 1 and N ľ 3 [ ].

Theorem . . Let u ∈ R 0 .

. Let N ľ 3. There exists some map v ∈ R 0 such that Jv = 0 in D (Ω) and

|v -u| p W s,p À ||Ju|| [Lip 0 (Ω)] * , ∀ 0 < s < N -1, 1 < p < ∞ s. t. sp = N -1. ( . )
. Let N = 2. There exists some map v ∈ R 0 such that Jv = 0 in D (Ω) and

|v -u| p W s,p À ||Ju|| [Lip 0 (Ω)] * , ∀ 0 < s ĺ 1, 1 ĺ p < ∞ s. t. sp = 1. ( . )
Elements of proof. For simplicity, we consider only the case where N ľ 3, and we prove the estimate ( . ) for a fixed couple (s, p). (For N = 2, see [ , Proposition . , Proposition . ].) The proof is very similar to the one of item in Theorem . . Let u ∈ R 0 and the sets U , A as in ( . ). We may assume that Ju = 0. Write, as in ( . ),

Ju = C N k (δ P k -δ N k )
, where the points P k , N k satisfy ( . )-( . ).

Step . Modification of u near its singularities. The purpose of this step is to obtain a new map, r u ∈ R 0 such that . ) near each of its singularities in Ω, r u satisfies the assumption (i) of Lemma . .

J r u = Ju in D (Ω), ( . ) ||r u -u|| p W s,p À ||Ju|| [Lip 0 (Ω)] * , (

( . )

This modification is performed in Lemma . . For this step, we require 0 < s < N -1 and we exclude the couple (s, p) = (N -1, 1). For the record, it is possible to extend the validity of Lemma . to this couple, if, for each singularity x ∈ A, we have deg(u, x) = 0.

Step . Dipole insertion, and conclusion. By Step , we may assume that ( . ) holds for u (instead of r u). We next construct a map v ∈ R 0 such that

Jv = 0 in D (Ω), ||v -u|| p W s,p À k |P k -N k |. ( .
)
The construction is performed using the procedure explained in Steps and of the proof of Theorem . , by inserting, at each singularity P k (respectively N k ) a dipole of degree -1 (respectively +1).

We complete the proof by noting that ( . ), ( . ), and ( . ) imply ( . ). Theorem . . Let N ľ 1, 0 < s ĺ 1, 1 ĺ p < ∞ be such that N -1 ĺ sp < N . Then, for u ∈ W s,p (Ω; S N -1 ), we have

u ∈ C ∞ (Ω; S N -1 ) W s,p ⇐⇒ Ju = 0.
Elements of proof. The implication " =⇒ " follows from Theorem . and the fact that, when u ∈ C ∞ (Ω; S N -1 ), we have Ju = 0. The remaining part of the proof is devoted to the reverse implication. In Step , we limit ourselves to the case 0 < s < 1, since we rely on our constructive proof of Theorem . for 0 < s < 1. However, as explained in [ ], we could have completed Step even for s = 1, by combining our argument with Bethuel's constructive proof of Theorem . when

s = 1 [ ].
Step . " ⇐= " holds when: (i) sp = N -1; (ii') 0 < s < N -1 when N ľ 3; (ii") 0 < s ĺ 1 when N = 2. By Theorem . , there exists a sequence (u i ) ⊂ R 0 such that u i → u in W s,p . By Theorem . and the assumption Ju = 0, we have Ju i → 0 in [Lip 0 (Ω)] * as i → ∞. By Theorem . , there exists a sequence (v i ) ⊂ R 0 such that Jv i = 0, ∀ i, and v i → u in W s,p . In order to complete this step, it remains to prove that

[v ∈ R 0 , Jv = 0] =⇒ v ∈ C ∞ (Ω; S N -1 ) W s,p .
( . )

In order to prove ( . ), we argue as in the proof of Lemma . . Since deg(v, x j ) = 0 near each singularity x j of v, we find that the restriction of v to a small sphere S δ (x j ) around x j is homotopic to a fixed constant a ∈ S N -1 , and then, for every µ > 0, we may construct, as in the proof of Lemma . , a smooth map r v : Ω → S N -1 such that ||r v -v|| W s,p ĺ µ and, near each x j , r v = a. This construction completes Step .

Step . " ⇐= " holds when 0 < s < 1 and N -1 < sp < N . (Sketch of proof.) We work with |x|:= ||x|| ∞ . Let Ω δ := {x ∈ Ω; d(x, ∂Ω > δ)}. We will prove that, for each δ > 0, u |Ω δ can be approximated in W s,p with maps in C ∞ (Ω δ ; S N -1 ). The same conclusion on Ω will then follow by a standard argument based on domain di feomorphisms. In order to simplify the formulas, we work in Ω rather than Ω δ , and then we may assume that u ∈ W s,p (U ; S N -1 ) and Ju = 0 in D (U ), where U is the larger domain {x ∈ R N ; d(x, Ω) < δ}. By Step , we have

u ∈ C ∞ (U ; S N -1 ) W σ,q , ∀ 0 < σ < N -1, 1 < q < ∞ s. t. σq = N -1. ( . )
Consider an extension of u, denoted f , to R N , such that f ∈ W s,p (R N ; R N ). (We do not claim that the extension is S N -1 -valued.) Let 0 < ε < δ/2. Using the notation in the proof of Theorem . , formula ( . ) holds for f (by Theorem . ). Moreover, as explained at the beginning of the Section . , for a. e. t ∈ Q ε , f satisfies ( . ) with j = N -1 and ( . ) with m = N -1. By

Step , there exists a sequence (u i ) ⊂ C ∞ (U ; S N -1 ) such that u i → u in W (N -1)/p,p (U ; S N -1 ). By slicing [ , Lemma . ], possibly a ter passing to a subsequence, still denoted (u i ), for a. e. t ∈ Q ε , we have

u i|C N -1,t,ε ∩U → u |C N -1,t,ε ∩U in W s,p .
( . )

Consider now a t ∈ Q ε such that ( . ), ( . ), and ( . ) hold, and any fixed cube C ∈ C N,t,ε contained in U . Let v i := u i|∂C , v := u |∂C . Note that v has a continuous representative (since ( . ) holds with j = N -1). By the above and a homotopy argument, we have

0 = lim i deg(u i , S ε (t)) = deg(v, S ε (t)), ( .
)
where the second equality follows from the embedding W s,p (∂C) → C 0 and the stability of the Brouwer degree under uniform convergence.

Consider now the smoothing process described in Section . : on the cubes C ∈ C N,t,ε contained in U and such that ( . ), ( . ), and ( . ) hold, we may approximate u with a (N -1)homogeneous map w such that its restriction on the boundary of each C is Lipschitz and (by the above stability argument) has zero degree. By Lemma . , we may approximate, in W s,p , w (and thus u) with Lipschitz S N -1 -valued maps. By an additional smoothing argument, we may approximate, in W s,p , u with maps in C ∞ (Ω; S N -1 ).

QED

For the record, let us note that the second step above has little to do with sphere-valued maps. It reveals a more general scheme that we formalize in the next statement.

Proposition . . Let:

(i) N ľ 2. (ii) 0 < s ĺ 1, 1 ĺ p < ∞, 1 ĺ j < N such that j < sp < j + 1. (iii) If j = N -1, Ω is any Lipschitz bounded domain. If j < N -1, we take Ω = (0, 1) N . Set σ := j/p < s. Let u ∈ W s,p (Ω; N ). Then u ∈ C ∞ (Ω; N ) W s,p ⇐⇒ u ∈ C ∞ (Ω; N ) W σ,p
. Sketch of proof. " =⇒ " is clear. For the reverse implication, we argue essentially as in Step above, and let U as there. First, we introduce an ad hoc notation. Let r

C N = r C t,N,ε := ∪{C ∈ C N ; C ⊂ U } and, for 0 ĺ j ĺ N -1, r C j := C j ∩ r C N . Let t ∈ Q ε be such that ( . )-( . ) hold and u i | r C j → u | r C j in W s,p , ∀ 0 ĺ j ĺ N -1. ( . )
By the stability argument leading to ( . ), for every face C ∈ r C j+1 , u |∂C : ∂C → N is null homotopic. By the smoothing process described in Section . and the multi-sequences Brezis-Lieb Lemma . , we may approximate in W s,p , on r C j , u | r C j with Lipschitz maps, null homotopic on each ∂C with C as above. We now invoke Lemma . and approximate, on each C, the homogeneous extension of u |∂C to C with a Lipschitz map. Then apply again Lemma . to obtain a global W s,p -approximation. To summarize, we have sketched the argument of the fact that, for a "generic" t ∈ Q ε , the homogeneous extension H j+1 of u | r C j to r C j+1 can be approximated, in W s,p , with Lipschitz maps. If j = N -1, then we found that u itself may be approximated with Lipschitz (and then smooth) N -valued maps, and we are done. When j < N -1, we use the fact that r C N is, up to an a fine transformation, a cube, and apply the singularities removing technique described in Section . to approximate, in W 1,q (and thus in W s,p , by Gagliardo-Nirenberg),

H N • • • • • H j+1 (u | r C j )
with Lipschitz N -valued maps and thus, finally, u with smooth N -valued maps. QED . .

Overview of the higher co-dimensional case

We let here N > k ľ 1 and consider maps u : Ω ⊂ R N → S k . The case where N = k + 1 corresponds to Sections . -. ; here, we rather focus on the case where N ľ k + 2. For the exposition in this section and beyond, we refer the reader to Alberti [ ] and[

. . Jacobian and singularities

We will consider a slightly di ferent route from the one in Section . . We start with the analytical definition of the distributional Jacobian (analogue of ( . )) rather than its geometric definition (analogue of ( . )). For

u ∈ W 1,k (Ω; S k ), set u ω k := k+1 j=1 (-1) j-1 u j du 1 ∧ . . . ∧ du j ∧ . . . ∧ du k+1 ∈ L 1 (Ω; Λ k ), ( . )
where ω k is the standard volume form ω k on S k (ω k corresponds to the choice N = k + 1 in ( . )). Note that, in particular, the definition ( . ) makes sense for u ∈ R N -k-1 . Then define

Ju := 1 k + 1 d(u ω k ) ∈ D (Ω; Λ k+1 ). ( . )
The above definitions are consistent with the ones in Section . . Let us note that, in the previous sections, it turned out to be more convenient to identify the N -form Ju with its density (a scalar distribution). The same situation occurs in any dimension: it will be more convenient to work with the (N -k -1)-form * Ju (where * stands for the Hodge operator) rather than with the (k + 1)-form Ju. In this perspective, and with the notation of the present section, we should have written, on the le t-hand side of ( . ), * Ju(ϕ) rather than Ju(ϕ).

We start with a fundamental example connecting Jacobians and singularities; see Jerrard and Soner [ , Section ] (and, also, Bousquet [ ]) for the first item, and Alberti, Baldo, and Orlandi [ , Theorem . ] for the second one.

Theorem . . . Let Γ be a smooth connected oriented (N -k -1)-submanifold Γ without boundary (in Ω). Let u ∈ W 1,k (Ω; S k ) ∩ C(Ω \ Γ). Then, with m := deg(u, Γ), we have * Ju = C k+1 m Γ in D (Ω; Λ N -k-1 ), ( . )
where C k+1 is the volume of the unit ball in R k+1 and Γ is identified, as usual, with an (Nk -1)-current.

. Given any connected Γ of the form Γ = r Γ ∩ Ω, with r Γ ⊂ R N a smooth closed oriented (N -k -1)-manifold, and any m ∈ Z, there exists u ∈ W 1,k (Ω; S k ) such that ( . ) holds.

Remark . . Some comments are in order concerning the statement of Theorem . . a) The requirement that Γ is oriented is important only when N -k -1 ľ 2. Indeed, points and curves are always orientable, but not, for example, surfaces in R 4 .

b) Given an oriented submanifold Γ in R N , we may always choose a coherent orientation of the normal spaces

: if {e 1 , . . . , e N -k-1 } is a direct basis of T x Γ, then a basis {e N -k , . . . , e N } of N x Γ is direct if {e 1 , . . . , e N } is a direct basis of R N .
c) The degree m = deg(u, Γ) is defined as follows. Let x ∈ Γ. Consider a (small) k-dimensional (hyper)sphere, denoted S ε (x), of radius ε, in the (geometric) normal (k + 1)-plane to Γ at x. The map u |Sε(x) : S ε (x) → S k is continuous. By a homotopy argument, its Brouwer degree, denoted here m, does not depend on x or (small) ε.

d) The meaning of ( . ) is the following:

ˆΩ dϕ ∧ (u ω k ) = (-1) N -k (k + 1)C k+1 m ˆΓ ϕ, ∀ ϕ ∈ Lip 0 (Ω; Λ N -k-1 ). ( . )
e) Item cannot be an exact converse to item , in the sense that the condition that Γ has no boundary in Ω is not su ficient for the existence of u as in item . Here is an example. Let Ω := B 10 (0) \ B 1 (0) and let Γ be the oriented segment from (0, 0, 1) to (0, 0, 10). Then there exists no u ∈ C(Ω \ Γ; S 1 ) such that deg(u, Γ) = 0 (and thus, in particular, for this Γ and for m = 1, the conclusion of item does not hold). Indeed, argue by contradiction, and consider some u ∈ C(Ω \ Γ; S 1 ), such that deg(u, Γ) = m = 0. We note that every circle

C ε (x) as in the definition of deg(u, Γ) is homotopic, in Ω \ Γ, to the circle x 2 1 + x 2 2 = 1 x 3 = -2
, which in turn is homotopic to a point. Via a homotopy argument, we find that m = 0 -a contradiction.

We next present a limitation of the use of the Jacobian as a "singularities detector" (see [ , Section . . ]). andH ∈ C 1 (S 3 ; S 2 ), then Ju = 0 in D (Ω). (Note that this u belongs to W 1,p (Ω; S 2 ), ∀ p < 4, by Lemma . a).) This implies that Ju does not detect lower dimensional "topological singularities," since H may carry a non-trivial Hopf degree.

Proposition . . Let u ∈ W 1,k (Ω; S k ) ∩ C(Ω \ Γ), where Γ ⊂ Ω is a closed set such that H N -k-1 (Γ) = 0. Then Ju = 0. Remark . . Proposition . implies that, if u(x) = H(x/|x|), where x ∈ Ω ⊂ R 4 , 0 ∈ Ω,
On the other hand, if H is topologically non-trivial, then, by the argument in Section . , u cannot be strongly approximated, in W 1,p (Ω; S 2 ), 3 ĺ p < 4, with smooth maps. This implies that the condition Ju = 0 is not su ficient for approximability with smooth maps.

. . The distributional Jacobian. Disintegration (slicing)

We present here the higher-dimensional counterpart of Section . . We assume that

0 < s < ∞, 1 ĺ p < ∞, k ĺ sp < k + 1. ( . )
The main result here is the following (see [ ]).

Theorem . . Assume ( . ). Then the map

R N -k-1 u J Þ -→ Ju ∈ D (Ω; Λ k+1 ) ( . )
has a unique extension by continuity, still denoted J, to W s,p (Ω; S k ).

In addition, Ju belongs to the space [Lip 0 (Ω; Λ N -k-1 )] * , the mapping W s,p (Ω; S k ) u → Ju ∈ [Lip 0 (Ω; Λ N -k-1 )] * is continuous, and we have the estimate

||Ju|| [Lip 0 (Ω;Λ N -k-1 )] * À |u| k/s W s,p .
( . )

The proof follows essentially Steps -of the proof of Theorem . .

We next connect the distributional Jacobian defined above with the definition in Theorem . . For simplicity, we let Ω = (0, 1) N . Set

I(N -k -1, N ) := {α ⊂ {1, . . . , N }; Card α = N -k -1}. For α ∈ I(N -k -1, N ), set α := {1, . . . , N } \ α ∈ I(k + 1, N ). Let ϕ ∈ C ∞ c (Ω; Λ N -k-1
). Then we may write

ϕ = α∈I(N -k-1,N ) ϕ α dx α = α∈I(N -k-1,N ) (ϕ α ) xα (x α ) dx α .
Here, dx α denotes the canonical (N -k -1)-form induced by the coordinates x j , j ∈ α, and

(ϕ α ) xα (x α ) := ϕ α (x α , x α ) belongs to C ∞ c ((0, 1) N -k-1 ; R) (for fixed x α ). Given u ∈ W s,p
(Ω; S k ), by slicing (Corollary . ), for a. e. x α ∈ (0, 1) N -k-1 the partial map (u α ) xα := x α → u(x α , x α ) belongs to W s,p ((0, 1) k+1 ; S k ). Assuming that s, p satisfy ( . ), for such x α the distributional Jacobian J(u α ) xα (or rather, as we have explained, * J(u α ) xα ) is welldefined (via Theorem . ) as an element of D (Ω).

We have the following disintegration result.

Proposition . . Assume ( . ). Let Ω = (0, 1) N and u ∈ W s,p (Ω; S k ). Then, with appropriate ε(α) ∈ {-1, 1} depending only on k, N , and

α ∈ I(N -k -1, N ), we have * Ju(ϕ) = α∈I(N -k-1,N ) ε(α) ˆ(0,1) N -k-1 * J(u α ) xα ((ϕ α ) xα ) dx α . ( . )
When s ľ 1, ( . ) follows from the Fubini theorem. The case where 0 < s < 1 is more delicate. See [ , Lemma . ] for a proof when s = 1/p and k = 1. (The argument there works in the general case.)

. . The range of the distributional Jacobian I will try not to appeal here to the language of geometric measure theory. For the same story (moderately) using this language, see [ , Chapter , Chapter ]. Let n := N -k ĺ N -1. Consider a C 1 oriented n-dimensional submanifold Σ of Ω, and a Borel set A ⊂ Σ such that H n (A) < ∞. Then A acts by integration on smooth compactly supported n-forms, through the formula

A(ζ) := ˆA ζ, ∀ ζ ∈ C ∞ c (Ω; Λ n ). ( . )
Note that the integral makes sense since A is oriented.

This allows us to identify A with a linear object (a distribution, or rather a current), and thus allow operations as (infinite) sums.

Consider the set

F n := T = j A j ; A j ⊂ Ω is a Borel subset of a C 1 oriented n-dimensional submanifold Σ j of Ω, j H n (A j ) < ∞ . ( . )
Given T ∈ F n (or, more generally, a distribution acting on smooth compactly supported n-forms), we define the boundary ∂T of T through the formula

∂T (ϕ) := T (dϕ), ∀ ϕ ∈ C ∞ c (Ω; Λ N -1 ). ( .
)
The terminology is justified by the fact that, when T is the integration over a compact oriented manifold with boundary, ∂T is the integration over the geometric boundary ∂T of T . (In this case, ( . ) is simply the Stokes theorem.)

We may now present the higher co-dimensional counterpart of Theorem . .

Theorem . . Assume ( . ).

. If u ∈ W s,p (Ω; S k ), then there exists some

T = j A j ∈ F N -k such that j H N -k (A j ) À |u| k/s W s,p , ( . ) * Ju = C k+1 ∂T in D (Ω; Λ N -k-1 ). ( . )
. Conversely, given T = j A j ∈ F N -k , there exists u : Ω → S k such that, for every s, p

satisfying sp = N -1: (i) u ∈ W s,p (Ω; S k ); (ii) ( . ) holds; (iii) |u| p W s,p À inf k H N -k ( r A k ); ∂ k r A k = ∂ j A j in D (Ω; Λ N -k-1 ) . ( . )
Already the fact that Theorem . is the special case of Theorem . with N = k + 1 is not obvious (but not di ficult to prove). When s = 1, item is due to Alberti, Baldo, and Orlandi [ , Theorem . ], who extended to general maps an argument relying on the co-area formula devised by Almgren, Browder, and Lieb [ ]. This main idea is illustrated, when u is su ficiently smooth, in [ , Section . ]. Item with s = 1 is also due to Alberti, Baldo, and Orlandi [ , Theorem . ]. It relies on a delicate dipole insertion technique, reminiscent of the one in Brezis, Coron, and Lieb [ ], but technically much more involved. The general case (arbitrary s, p) was obtained in [ ].

. . Characterization of the closure of smooth maps

We have the following counterpart of Theorem . (with the same references as for Theorem . ).

Theorem . . Assume that 0 < s ĺ 1, 1 ĺ p < ∞, k ĺ sp < k + 1. Moreover, when k < sp < k + 1 and N > k + 1, assume that Ω = (0, 1) N . Then, for u ∈ W s,p (Ω; S k ), we have

u ∈ C ∞ (Ω; S k ) W s,p ⇐⇒ Ju = 0.

Appendix # . Standard & less standard properties of Sobolev spaces

We present here, mostly without proofs, some of the basic properties of Sobolev maps that we use in the main text. For the full proofs, some useful general references are Triebel [ , ] In what follows, Ω ⊂ R N is a Lipschitz bounded domain. Occasionally, it could be R N or a half space. Unless specified otherwise, the Sobolev spaces W s,p and the corresponding norms are considered with respect to Ω.

Contents

Slicing and characterization via di ferences

For simplicity, we only consider the case of one-dimensional slices, i. e., given a map u = u(x 1 , . . . , x N ), we connect its regularity with the one of its one-dimensional slices u(x 1 , . . . , x k-1 , •, x k+1 , . . . , x N ), k = 1, . . . , N , but similar results are available for -dimensional slices. Theorem . . Let s > 0 be non-integer. Then

||u|| p W s,p ∼ ||u|| p p + N k=1 ˆ1 -1 ˆˆ|δ M te k u(x)| p |t| 1+sp dx ˙dt. ( . )
Here:

. t is a one-dimensional variable.

. M is any integer satisfying M > s.

. The integral in x is computed over the set {x ∈ Ω; [x, x + M te k ] ⊂ Ω}.

. δ h u(x) := u(x + h) -u(x), and

δ M h := δ h • . . . • δ h l jh n M times .
For a proof, see [ , Section . . ], [ , Section . . ]. An immediate consequence of ( . ) is the following Corollary . . Let s > 0 be non-integer and 1 ĺ p < ∞. Then

u p W s,p (R N ) ∼ N k=1 ˆ u(x 1 , . . . , x k-1 , •, x k+1 , . . . , x N ) p W s,p (R) dx k . ( . )
In particular, for a.e. (x 1 , . . . , x N -1 ) ∈ R N -1 , we have u(x 1 , . . . , x N -1 , •) ∈ W s,p (R).

Here, dx j := dx 1 . . . dx k-1 dx k+1 . . . dx N .

A straightforward consequence of ( . ) and of the "converse" to the dominated convergence theorem is the following Fubini type convergence result.

Corollary . . Let s > 0 be a non-integer and 1 ĺ p < ∞. Assume that u j → u in W s,p (R N ). Then, possibly up to a subsequence, we have

u j (x 1 , . . . , x N -1 , •) → u(x 1 , . . . , x N -1 , •) in W s,p (R) for a.e. (x 1 , . . . , x N -1 ) ∈ R N -1 .
A variant of ( . ) holds for both fractional and integer Sobolev spaces. Let u : R N → R. If ω ∈ S N -1 , let ω ⊥ denote the hyperplane orthogonal to ω, and consider the partial functions ω ⊥ x → u x ω , with u x ω (t) := u(x + t ω), ∀ t ∈ R. Then we have the following Proposition . . Let s ľ 0 and 1 ĺ p < ∞. Let u : R N → R. Then

u p W s,p (R N ) ∼ ˆSN-1 ˆω⊥ u x ω p W s,p (R) dx dH N -1 (ω). ( . )
For a proof, see [ , Lemma ].

. .

Sobolev embeddings

Optimal Sobolev embeddings are of the form W s,p (Ω) → W r,q (Ω), where s, r, p, q satisfy s > r ľ 0, 1 ĺ p < q ĺ ∞, s -N p = r -N q .

( . )

Note that we allow the value q = ∞.

The following result incorporates the classical Sobolev, Morrey, and Gagliardo-Nirenberg embeddings.

Theorem . . Let s, r, p, q, N satisfy ( . ). Then we have

W s,p (Ω) → W r,q (Ω) ( . )
with the following exceptions, when ( . ) does not hold.

(a) When N = 1, s is an integer ľ 1, p = 1, 1 < q < ∞, and r = s -1 + 1/q, ( . ) we have W s,1 (Ω) → W s-1+1/q,q (Ω).

( . )

In particular, we have W 1,1 ((0, 1)) → W 1/q,q ((0, 1)), 1 < q < ∞.

( . ) . .

(b) When N ľ 1, 1 < p < ∞, q = ∞,

Gagliardo-Nirenberg inequalities

Consider the estimate

u W s,p À u θ W s 1 ,p 1 u 1-θ W s 2 ,p 2 , ∀ u ∈ W s 1 ,p 1 ∩ W s 2 ,p 2 , ( .
)
where s 1 , s 2 , s ľ 0 and 1 ĺ p 1 , p 2 , p ĺ ∞ are related by

   s = θs 1 + (1 -θ)s 2 1 p = θ p 1 + 1 -θ p 2
for some θ ∈ (0, 1).

( . )

With no loss of generality, we may assume that

s 1 ĺ s 2 . ( . )
The following condition plays an essential role in the validity of ( . ):

s 2 is an integer ľ 1, p 2 = 1 and s 2 -s 1 ĺ 1 - 1 p 1 . ( . )
(The latter condition can also be written in the more symmetric form s .1 -1/p 1 ľ s 2 -1/p 2 .)

We have the following result [ ] incorporating the classical Gagliardo-Nirenberg inequalities [ , ].

Theorem . . Let s, s 1 , s 2 , p, p 1 , and p 2 satisfy ( . ) and ( . ). Then, ( . ) holds if and only if ( . ) fails.

More precisely, we have . If ( . ) fails, then, for every θ ∈ (0, 1),

u W s,p À u θ W s 1 ,p 1 u 1-θ W s 2 ,p 2 , ∀ u ∈ W s 1 ,p 1 ∩ W s 2 ,p 2 . ( . )
Moreover, if s 1 < s < s 2 , then we have (in a bounded domain) the estimate ) . If ( . ) holds, there exists some u ∈ W s 1 ,p 1 ∩ W s 2 ,p 2 such that u ∈ W s,p , ∀ θ ∈ (0, 1).

|u| W s,p À u θ W s 1 ,p 1 |u| 1-θ W s 2 ,p 2 , ∀ u ∈ W s 1 ,p 1 ∩ W s 2 ,p 2 . ( .
Here is the special case of the above theorem we use the most o ten in this text.

Corollary . . The embedding

W s,p ∩ L ∞ → W θs,p/θ , ∀ 0 < θ < 1,
and the corresponding estimate ( . ) hold, ∀ s > 0, ∀ 1 ĺ p ĺ ∞, except when s = p = 1.

. .

Trace theories

We consider, to simplify the presentation, only maps u : R N → R. In order to obtain the corresponding results on domains, either we extend maps from domains to R N , or we work directly in domains and then deform the domain {(x, ε); 0 < ε < ε 0 , x ∈ Ω, d(x, ∂Ω) > ε} to the cylinder Ω × (0, ε 0 ) (as explained in the proof of Proposition . ). The classical results presented here are due to several authors, including Gagliardo [ ] and Uspenskiȋ [ ]. For the proofs, see, e. g., [ , Section ], [ ], [ , Lemma . ].

Let u : R N → R, ρ be a standard mollifier, and set U (x, ε)

:= u * ρ ε (x), ∀ x ∈ R N , ∀ ε > 0.
We first present trace theory in weighted Sobolev spaces.

Theorem . . (Inverse trace theory in weighted Sobolev spaces) Let 0 < s < m, with s is noninteger and m integer. We have, with C = C(m, s, p, N ),

|α|=m ˆRN ˆ∞ 0 ε p(m-s)-1 |∂ α U (x, ε)| p dεdx ĺ C|u| p W s,p , ∀ u : R N → R. ( . )
Theorem . . (Direct trace theory in weighted Sobolev spaces) Let 0 < s < m, with s is noninteger and m integer.

Let V ∈ C ∞ (R N × (0, ∞); R). We have, with C = C(m, s, p, N ), ∀ 0 < ε < 1/2, ||V (•, ε)|| p W s,p ĺ C |α|=m ˆRN ˆ∞ 0 ε p(m-s)-1 |∂ α V (x, ε)| p dεdx + C||V || p L p (R N ×(0,1)) . ( . )
Moreover, if the right-hand side of ( . ) is finite, then the limit u := lim ε→0 V (•, ε) exists in W s,p and satisfies

||u|| p W s,p ĺ C |α|=m ˆRN ˆ∞ 0 ε p(m-s)-1 |∂ α V (x, ε)| p dεdx + C||V || p L p (R N ×(0,1)) . ( . )
We next present trace theory in fractional Sobolev spaces. (The two theories coincide when s + 1/p is an integer.)

Let V ∈ C ∞ (R N × (0, ∞); R). We have, with C = C(s, p, N ), ∀ 0 < ε < 1/2, ||V (•, ε)|| W s,p ĺ C|V | W s+1/p,p +C||V || L p (R N ×(0,1)) . ( . )
Moreover, if the right-hand side of ( . ) is finite, then the limit u := lim ε→0 V (•, ε) exists in W s,p and satisfies

||u|| W s,p ĺ C|V | W s+1/p,p +C||V || L p (R N ×(0,1)) .
( . )

. .

Superposition operators

Given Φ : R → R and u : Ω → R (or Φ : R M → R K and u : Ω → R M ), set

F (u) := Φ • u. ( . )
Here are the results on the continuity of F we rely on.

Theorem . . Assume that Φ is Lipschitz. Let 0 < s ĺ 1, 1 ĺ p < ∞. Then F (given by ( . )) is continuous from W s,p in itself.

The non-trivial part of the above result is continuity. For a proof when s = 1, see Marcus and Mizel [ ]. For the case 0 < s < 1, see, e. g., [ , proof of ( . )]).

Theorem . . Let s > 1 and 1 ĺ p < ∞. Let m denote the first integer ľ s. Let Φ ∈ C m . Then F (given by ( . )) maps W s,p ∩ L ∞ into W s,p and is continuous in the following sense:

if u n → u in W s,p and u n L ∞ ĺ C, then Φ(u n ) → Φ(u) in W s,p .
For an elementary proof of the above result for arbitrary s, see Escobedo [ ].

Theorem . . Let s > 1 and let 1 ĺ p < ∞. Let m be the least integer ľ s. Let Φ ∈ C m (R M ; R K ) have bounded derivatives of order ĺ m. Then, for every u ∈ W s,p ∩ W 1,sp (Ω; R M ), we have F (u) ∈ W s,p (Ω; R K ). In addition, F (given by ( . )) is continuous from W s,p ∩ W 1,sp to W s,p .

For a proof, see [ ]. (The result there is stated for p > 1, but exactly the same proof applies when p = 1.) See also Maz'ya and Shaposhnikova [ ]. Corollary . . Let 0 < s < ∞, 1 ĺ p < ∞. If either s ĺ 1 or sp ľ N , then ϕ → e ıϕ acts continuously from W s,p (Ω; R) to W s,p (Ω; S 1 ). Proof. When s ĺ 1, this is a special case of Theorem . . When s > 1, by assumption we have sp ľ N , and in this case we rely on Theorem . and on the Sobolev embedding W s,p → W 1,sp .

QED

. . Products

The most used product property of the Sobolev spaces is that W s,p ∩ L ∞ is an algebra, in the following sense.

Lemma . . Let s > 0, 1 ĺ p < ∞. If u, v ∈ W s,p ∩ L ∞ , then uv ∈ W s,p and uv W s,p ĺ C( u L ∞ v W s,p + v L ∞ u W s,p ). ( . )
In addition, the map (u, v) → uv is continuous in the following sense: if u n → u and v n → v in W s,p , and if

u n L ∞ , v n L ∞ ĺ C, then u n v n → uv in W s,p .
For a proof, see, e. g., Runst and Sickel [ ].

When s > 1, the above result can be strengthened as follows.

Lemma

. . Let 1 < s < ∞, 1 ĺ p < ∞. Let u, v ∈ W s,p ∩ L ∞ . Then uDv ∈ W s-1,p .
For a proof, see, e. g., [ , Appendix . ].

A refinement of Lemma . is provided by the following result [ , Lemma . ]. † Lemma . . A couple (s, p), with s > 0 and 1 ĺ p < ∞, is regular if either p > 1 or p = 1 and s is not an integer.

Let (s 1 , p 1 ), (s 2 , p 2 ) be two regular couples such that s 1 > s 2 and s 1 p 1 > s 2 p 2 . Let 1 < r < ∞ be defined by

1 r = 1 p 2 - s 2 s 1 p 1 . If f ∈ W s 1 ,p 1 ∩ L ∞ (Ω) and g ∈ L s 2 ,p 2 ∩ L r (Ω), then f g ∈ W s 2 ,p 2 (Ω).

. . Gluing

We have the following straightforward versions of the Brezis-Lieb lemma [ ].

Lemma . . Let f ∈ W s,p (Ω). Consider a bounded sequence of maps f j ∈ W s,p (Ω) such that

lim j→∞ |∪ kľj supp f k |= 0. Then |f + f j | p W s,p = |f | p W s,p +|f j | p W s,p +o(1) as j → ∞, ||f + f j || p p = ||f || p p + ||f j || p p + o(1) as j → ∞. Lemma . . (Multi-sequences Brezis-Lieb lemma) Let f ∈ W s,p (Ω). Consider m bounded se- quences (f j,1 ), . . . (f j,m ) ⊂ W s,p (Ω) such that lim j→∞ |∪ kľj supp f k,i |= 0, ∀ i, |f j,i | p W s,p ĺ C i , ∀ j, ∀ i, ||f j,i || p p ĺ D i , ∀ j, ∀ i.
Then there exist j 1 , . . . , j m such that

|f + f j 1 ,1 + • • • + f jm,m | p W s,p ĺ |f | p W s,p +2(C 1 + . . . + C m ), |f j 1 ,1 + • • • + f jm,m | p W s,p ĺ |f | p W s,p +2(C 1 + . . . + C m ), ||f + f j 1 ,1 + • • • + f jm,m || p p ĺ ||f || p p + 2(D 1 + . . . + D m ), ||f j 1 ,1 + • • • + f jm,m || p p ĺ 2(D 1 + . . . + D m ).
In view of the applications, let us note that, in Lemma . , the indices j k may be chosen inductively with respect to k. † The result in [ ] is stated for 1 < p 1 , p 2 < ∞, but the proof there is valid for all regular couples. . .

Quantitative suboptimal Sobolev embeddings

In view of the applications we consider, we work here in the unit ball Ω (for some norm | |in R N ) and with less than one derivative, but what follows can be adapted to any domain and to higher order spaces.

Consider parameters satisfying

0 ĺ α < s < 1, 1 ĺ p < ∞, 1 ĺ q ĺ ∞, α - N q < s - N p , p ĺ t ĺ ∞. ( . ) When α = 0, set |u| W 0,q (Br(x)) := u - Br(x) u L q (Br(x))
.

Lemma . . Assume ( . ). Set

β := s -α -N ˆ1 p - 1 q ˙> 0, ( . )
U := {(x, r); x ∈ Ω, r > 0, B r (x) ⊂ Ω}. ( . )
Then (with the obvious modification when t = ∞)

˜ˆU |u| t W α,q (Br(x)) r βt dxdr r N +1 ¸1/t À |u| W s,p , ∀ u : Ω → R. ( . )
Proof. Let α < σ ĺ s be such that

γ := σ -α -N ˆ1 p - 1 q ˙> 0.
By Sobolev's embedding, Poincaré's inequality, and scaling, we have

|u| W α,q (Br(x)) À r γ |u| W σ,p (Br(x)) , ∀ (x, r) ∈ U. ( . )
The choice σ = s yields ( . ) for t = ∞. It thus su fices to obtain ( . ) when t = p; the general case will follow via Hölder's inequality. For this purpose, we note that, with α < σ < s as above and δ := β -γ = s -σ > 0, we have (using ( . ))

ˆU |u| p W α,q (Br(x)) r βp dxdr r N +1 À ˆU 1 r δp+N +1 ¨Br(x) |u(w) -u(z)| p |w -z| N +σp dwdz dxdr = ¨Ω2 |u(w) -u(z)| p |w -z| N +σp ¨{(x,r)∈U; z,w∈Br(x)} 1 r δp+N +1 dxdr dwdz ĺ ¨Ω2 |u(w) -u(z)| p |w -z| N +σp ˆ∞ |w-z|/2 ˆBr(w) dx 1 r δp+N +1 dr dwdz ∼ ¨Ω2 |u(w) -u(z)| p |w -z| N +σp ˆ∞ |w-z|/2 1 r δp+1 dr dwdz ∼ ¨Ω2 |u(w) -u(z)| p |w -z| N +σp+δp dwdz = |u| p W s,p .

QED

We will use the following special case of Lemma . [ , Section . ].

Corollary . . Assume that 0 < s < 1, 1 ĺ p < ∞, and sp > 1. Then

ˆ1 0 ˆ1 0 |u| p W 0,∞ ((x,y)) |y -x| 1+sp dxdy À |u| p W s,p , ∀ u : (0, 1) → R. ( . )
Proof. Setting r := |y -x|/2, z := (y + x)/2, we see that the le t-hand side I of ( . ) satisfies

I À ˆ1 0 ˆ{r; Br(z)⊂(0,1)} |u| p W 0,∞ (Br(z))
r 1+sp drdz.

We conclude via ( . ) applied with α = 0, q = ∞, t = p, and N = 1.

QED

. . Martingales and Sobolev spaces

The framework is the one of Step of the proof of Theorem . . We work in Ω = [0, 1) N and with the || || ∞ norm, denoted for simplicity | |. For k ľ 0, let P k denote the collection of dyadic cubes of size 2 -k in Ω. Q k denotes a generic cube in P k , and, for x ∈ Ω, Q k (x) is the only cube in P k containing x. We let F k denote the set of the (step) functions constant on each

Q k . If u : Ω → R , let E k (u) ∈ F k be defined by E k (u)(x) := Q k (x)
u.

For the next result, see [ , Proof of Theorem A. , Step ].

Lemma . . Let 0 < s < 1. Then, for each f ∈ L p (Ω; R ), kľ0 2 spk ||f -E k (f )|| p p À |f | p W s,p .
We next present a variant of [ , Theorem A. ] adapted to our purposes.

Lemma . . Let 0 < s < 1 and 1 ĺ p < ∞ be such that sp < 1. Let Φ : Ω 2 → [0, ∞) be measurable and let, for each k ľ 1,

g k ∈ F k , g k ľ 0. If [x, y ∈ Q j ] =⇒ Φ(x, y) ĺ k>j (g k (x) + g k (y)), ∀ j ľ 0, ∀ Q j ∈ P j , (
(x) = Q k (y)}, t(x, y) := min{k; |x -y|> 2 -t }.
We note the following obvious facts: (i) s and t are symmetric; (ii) if k > s(x, y), then y ∈ Q k (x); (iii) t(x, y) > s(x, y); (iv) 2 -t(x,y) < |x -y|ĺ 2 1-t(x,y) .

In view of ( . ), in order to obtain ( . ) it su fices to prove the inequality ˆΩ ˆΩ ˆ k>s(x,y)

g k (x) ˙p dxdy |x -y| N +sp À K := kľ1 2 spk ||g k || p p . ( . )
Since, by Hölder's inequality, we have

ˆ k>s(x,y) (g k (x)) ˙p À kľt(x,y) (k -t(x, y) + 1) p g p k (x) + s(x,y)<k<t(x,y) (t(x, y) -k) p g p k (x),
it su fices to establish the inequalities

I := ˆΩ ˆΩ kľt(x,y) (k -t(x, y) + 1) p g p k (x) dxdy |x -y| N +sp À K, ( . )
J := ˆΩ ˆΩ s(x,y)<k<t(x,y) (t(x, y) -k) p g p k (x) dxdy |x -y| N +sp À K.
( . )

Step . ( . ) holds for 0 < s < 1 and measurable g k 's (without the assumptions sp < 1 or g k ∈ F k ). We have

I = kľ1 1ĺjĺk (k -j + 1) p ˆΩ g p k (x) ˆ{y∈Ω; t(x,y)=j} dy |x -y| N +sp dx À kľ1 1ĺjĺk (k -j + 1) p ˆΩ g p k (x) 2 spj dx À kľ1 2 skp ˆΩ g p k (x) dx = K.
Step . An auxiliary estimate. Let Q be a(ny) cube of size 2 -k in R N . If sp < 1, then

I k := j>k (j -k) p ¨{(x,y); x∈Q, y ∈Q; t(x,y)=j} dxdy |x -y| N +sp À 2 (sp-N )k . ( . )
Indeed, the le t-hand side of ( . ) does not depend on the center of the cube, and, by a scaling argument, we have I k = 2 (sp-N )k I 0 . It therefore su fices to prove that, with Q := (-1/2, 1/2) N , we have

I 0 = j>0 j p ¨{(x,y); x∈Q, y ∈Q; t(x,y)=j} dxdy |x -y| N +sp < ∞.
( . )

For this purpose, let us note that

|{x ∈ Q; |x|ľ 2 -1 -ε}|À ε, ∀ ε > 0, ( . ) [x ∈ Q, y ∈ Q, t(x, y) = j] =⇒ [2 -j < |x -y|ĺ 2 1-j and |x|ľ 1/2 -2 1-j ].
( . )

Combining ( . )-( . ) and using, at the end, the assumption sp < 1, we find that

I 0 ĺ j>0 j p ¨{(x,y); x∈Q, |x|ľ1/2-2 1-j , 2 -j <|x-y|ĺ2 1-j } dxdy |x -y| N +sp À j>0 j p ˆ{x; x∈Q, |x|ľ1/2-2 1-j } 2 spj dx À j>0 j p 2 spj 2 1-j = 2 j>0 j p 2 -(1-sp)j < ∞.
Step . ( . ) holds when sp < 1 and

g k ∈ F k . For Q k ∈ P k , let g k (Q k ) denote the value of g k on Q k . For k > s(x, y), we have y ∈ Q k (x).
Combining this observation with ( . ), we find that

J = kľ1 Q k ∈P k j>k (j -k) p g p k (Q k ) ¨{(x,y); x∈Q k , y ∈Q k ; t(x,y)=j} dxdy |x -y| N +sp À kľ1 Q k ∈P k g p k (Q k ) 2 (sp-N )k = kľ1 2 spk ||g k || p p = K. QED . .

Adapted trace theory

The following result is presented, with a sketch of proof, in Chiron [ , Section . ]. In the statement below, we impose an extra smoothness assumption on ζ that makes the arguments in [ ] essentially complete.

Lemma . . Let 0 < s < 1. Let k ľ 1 be the smallest integer such that s + k/p ľ 1. Let ζ ∈ W s+k/p,p ((0, 1) N +k ; E ) ∩ C ∞ . Then, for every x ∈ (0, 1) k , ||ζ(•, x )|| W s,p ((0,1) N ;E ) À ||ζ|| W s+k/p,p ((0,1) N +k ) .

. . Homotopy and VMO

We consider two closed embedded Riemannian manifolds, M ⊂ R k and N ⊂ R .

Lemma . . Let u j , u ∈ C(M ; N ) be such that u j → u in BMO and L 1 . Then, for large j, u j and u are homotopic in C(M ; N ).

Proof. Given a continuous map v : M → R , set v 0 := v and, for small ε > 0,

v ε (x) := Bε(x)
v(y) dy.

By the proof of Lemma . , we have

d(v ε (x), N ) ĺ sup y∈M Bε(y) Bε(y) |v(w) -v(z)| dwdz. ( . )
Let δ > 0 be such that the nearest projection on N is continuous on the δ-neighborhood N δ of N . By ( . ) and the continuity of u, there exists some ε 0 such that

d(u ε (x), N ) < δ/2, ∀ x ∈ M , ∀ ε ĺ ε 0 . ( . )
On the other hand, since u j → u in BMO, there exists some j 0 such that

sup y∈M Bε(y) Bε(y) |v(w) -v(z)| dwdz < δ/2, ∀ j ľ j 0 , ∀ y ∈ M , ∀ ε ĺ ε 0 . ( . )
Combining ( . )-( . ), we find that, for j ľ j 0 and 0 < ε ĺ ε 0 , (u j ) ε and u ε take values into N δ .

Let Π : N δ → N be the nearest point projection. Then, clearly, the map

[0, ε 0 ] ε → Π • [(u j ) ε ] ∈ C(M ; N )
is continuous, and therefore u j and Π • [(u j ) ε 0 ] are homotopic. A similar conclusion holds for u.

On the other hand, since

u j → u in L 1 , we clearly have Π • [(u j ) ε 0 ] → Π • [u ε 0 ]
uniformly, and thus, for large j, we have

u j ∼ Π • [(u j ) ε 0 ] ∼ Π • [u ε 0 ] ∼ u.

. . Characteristic functions

Lemma . . Let 0 < s < 1 and 1 ĺ p < ∞ be such that sp < 1. Let Q := (0, 1) N and Ω := (0, 1) N -1 × (-1, 1). Then ϕ N := χ Q belongs to W s,p (Ω).

Proof. In view of Theorem . (applied with M = 1), the conclusion of the lemma is equivalent to the straightforward fact that ˆ0 -1 ˆ1 0 1 |x -y| 1+σ dxdy < ∞, ∀ 0 < σ < 1.

( . )

Lemma . . Let s > 0 and 1 ĺ p < ∞. Let ω be a non-empty smooth relatively compact domain such that ω ⊂ Ω. Then ϕ := χ ω ∈ W s,p (Ω) ⇐⇒ sp < 1.

Proof. If we straighten the coordinates around a point of ∂ω, we find that, up to a constant factor, the le t-hand side of ( . ), with σ := sp, is a lower bound for |ϕ| p W s,p . When sp ľ 1, this implies that ϕ ∈ W s,p . Assume next that sp < 1. Let (U j ) be a finite covering of Ω such that, for each j, we have either ϕ = 0 in U j , or there exists a bi-Lipschitz di feomorphism Φ j of (0, 1) N -1 × (-1, 1) onto U j such that ϕ • Φ j = ϕ N (with ϕ N as in the previous lemma). For any such j, we have ϕ • Φ j ∈ W s,p ((0, 1) N -1 × (-1, 1)) (and thus ϕ ∈ W s,p in U j ). Finally, we have ϕ ∈ W s,p (U j ) for each j, which implies that ϕ ∈ W s,p (Ω). ( . )

Proof. If s is an integer, the conclusion is clear. We may therefore assume that s is not an integer and that α < N (the latter condition is equivalent to u ∈ L 1 ). We rely on Theorem . . Before going further, let us note that, since the specific ϕ we consider is smooth outside the origin, we have ϕ ∈ W s,p if and only if the double integral in ( . ) considered over the larger set [-1, 1] × Ω is convergent.

We use ( . ) with an integer M satisfying, in addition to M > s, the condition (α + M )p > N.

( . )

For simplicity, we drop the subscript j in e j . Using the homogeneity of ϕ, we find that δ M te ϕ(x) = 1 |t| α δ M e ϕ(x/t). Hence,

I := ˆ1 -1 ˆB1 (0)
|δ M te ϕ(x)| p t 1+sp dxdt = 2 ˆ1 0 ˆB1/t (0) t N -(α+s)p-1 |δ M e ϕ(y)| p dydt. ( . )

" ⇐= " Assume that (α + s)p < N . In this case, we show that I < ∞. Indeed, we have Here,we use the fact that ˆB1 (0) |δ M e ϕ(y)| p dy > 0 (since ϕ is not a polynomial in the e direction). QED Remark . . For a di ferent approach, see the proof below of Lemma . b).

. . Homogeneous maps

By repeating the proof of Lemma . , we obtain the following result. Then, u ∈ W s,p (B 1 (0)) if and only if sp < k.

c) Even more generally, let v : S k-1 → R be a smooth non constant map and

u : R N → R k , u(x 1 , . . . , x N ) := v ˆ(x 1 , . . . , x k ) |(x 1 , . . . , x k )| ˙, ∀ x ∈ R N \ ({0} × R N -k ).
Then, u ∈ W s,p (B 1 (0)) if and only if sp < k. In particular, Lemma . a) holds.

A more general result is the following Proof. We consider only the more complicated case where d ĺ N -2. Without loss of generality, we may assume that sp > N -d -1. Furthermore, we may also suppose that s > N -d -1, and thus N -d -1 < s < N -d (since sp < N -d). Indeed, if s ĺ N -d -1, fix any θ ∈ (0, 1) such that

1 p < θ < s N -d -1 ĺ 1,
and let r := s/θ > N -d -1, and q := pθ > 1. From the Gagliardo-Nirenberg embedding W r,q ∩ L ∞ → W s,p (see Corollary . ), and the fact that u |Ω ∈ L ∞ (by ( . ) with = 0), it su fices to prove that u |Ω ∈ W r,q . Thus, as claimed, it su fices to prove the lemma under the extra assumption s > N -d -1, which implies that N -d -1 < s < N -d.

Let m := N -d -1 and write s = m + σ, with 0 < σ < 1. The assumption sp < N -d reads (m + σ)p < N -d.

We set d(x) := dist (x, S). Since S is d-dimensional, we find that ˆΩ 1

d q (x) dx < ∞, ∀ 1 ĺ q < N -d. ( . )
We next invoke the following well-known result. Remark . . The conclusion of Lemma . is still valid under the weaker assumption that S is a finite union of d-dimensional submanifolds. This has the following important (for us) consequence: the class R defined in ( . ) satisfies R → W r,q (Ω; N ), ∀ r, q such that rq ĺ N --1.

Remark . . When s = 1, the above result (or versions of it) can be proved using more elementary arguments; see, e. g., [ , Lemma . ] for the following version of Lemma . .

Lemma .

. Let 1 ĺ r < ∞. Assume that N ľ 2 and let U ⊂ R N be an open set. Let K be a closed subset of U such that H N -1 (K) = 0. Let u ∈ W 1,1 loc (U \ K) be such that ´U\K |∇u| r < ∞. Then u ∈ W 1,r loc (U ) and the Sobolev gradient of u is the Sobolev gradient of u |U \K .

Proof of Lemma . b).

" ⇐= " Case s < 1. By Lemma . , we have ψ ∈ W 1,sp+ε , and thus also u ∈ W 1,sp+ε , for small ε > 0. By Corollary . , we find that u ∈ W s,p .

Case s = 1. The conclusion follows directly from Lemma . .

Case s > 1. By Lemma . , we have ϕ ∈ W s,p ∩ W 1,sp . We conclude via Theorem . . " =⇒ " Case s = 1. We have |∇u|= |∇ϕ|∼ |x| α-1 in Ω \ {0}, and the conclusion is clear.

Case s > 1. By Corollary . , we have u ∈ W 1,sp , and thus (as in the case s = 1) we find that (α + 1)sp < N . On the other hand, by di ferentiating the equality u = e ıϕ , we find that Dϕ = -ıuDu. Lemma . implies that Dϕ ∈ W s-1,p , and thus ϕ ∈ W s,p . Lemma . implies the second condition, (α + s)p < N .

Case 0 < s < 1. This case is more involved. Setting I(r 1 , r 2 , β) := ˆS(0,r 1 ) ˆS(0,r 2 ) dH (x)dH (y) |x -y| β , ( . )

we have, with β := N + sp, using the changes of variables t = r -α 1 , τ = r -α 2 , Lemma . below, and the fact that N -β + p > 0, |u| p W s,p =2 ˆ1 0 ˆr1 0 e ı/r α 2 -e ı/r α 1 p I(r 1 , r 2 , β) dr 2 dr 1 ∼ ˆ∞ 1 ˆ∞ t |e ıτ -e ıt | p I(t -1/α , τ -1/α , β)τ -1/α-1 t -1/α-1 dτ dt Á ˆ∞ 1 ˆt+1 t |e ıτ -e ıt | p t -(N -1)/α (t -1/α -τ -1/α ) N -β-1 τ -1/α-1 t -1/α-1 dτ dt ∼ ˆ∞ 1 ˆt+1 t (τ -t) p t -2-(N +1)/α (t -1/α -τ -1/α ) N -β-1 dτ dt Á ˆ∞ 1 ˆt+1 t (τ -t) p t -2-(N +1)/α (τ -t) N -β-1 t -(N -β-1)(1/α+1) dτ dt = ˆ∞ 1 ˆt+1 t (τ -t) N -β+p-1 t -(N -β+1)-(2N -β)/α dτ dt

Á ˆ∞ 1 t -(N -β+1)-(2N -β)/α dt.
Therefore, the exponent of t in the last integral above has to be < -1, and this amounts to (α + 1)sp < N . QED Lemma . . Fix β > 0. For r 2 < r 1 ĺ 2r 1 , we have

I(r 1 , r 2 , β) Á r N -1 1 (r 1 -r 2 ) N -β-1 .
Proof. Write r 1 = (1 + t)r 2 , with 0 < t ĺ 1. By scaling and invariance with respect to isometries, we have Write x = (x , x N ). We have, using the facts that t ĺ 1 and 1 -? 1 -s 2 ĺ t when 0 ĺ s ĺ t ĺ 1, and the change of variable x = ty ,

I(r 1 , r 2 , β) = σ N r N -1 1 r N -β
J(t, β) ľ ˆ|x |ĺ1 dx |(x , 1 + t - ? 1 -x 2 )| β ľ ˆ|x |ĺt dx |(x , 1 + t - ? 1 -x 2 )| β ľ ˆ|x |ĺt dx |(x , 2t)| β = t N -β-1
ˆ|y |ĺ1 dy |(y , 2)| β , so that, by ( . ),

I(r 1 , r 2 , β) Á r N -1 1 r N -β-1 2 t N -β-1 = r N -1 1 (r 1 -r 2 ) N -β-1 . QED . .

Gluing maps

Lemma . . Let π : E → N be a covering. Fix some z ∈ N and let J be at most countable such that π -1 ({z}) = {t j ; j ∈ J}. Then there exist: points x j ∈ Ω, radii r j > 0, j ∈ J, and maps ζ j : B 2r j (x j ) → E such that:

(i) The balls B 3r j (x j ) are mutually disjoint and contained in Ω.

(ii) ζ j ∈ C ∞ (B 2r j (x j ) \ {x j }).

(iii) ζ j = t j in B 2r j (x j ) \ B r j (x j ).

(iv) ζ j ∈ W s,p (B 2r j (x j )).

(v) The map u := π • ζ j , in B 2r j (x j ) z, in Ω \ ∪ j B 2r j (x j ) belongs to W s,p .

Proof. We consider only the non-compact case where J = {1, 2, . . .}. We will construct inductively a sequence of maps (u j ) such that its weak limits has all the required properties. Start with u 0 :≡ z. Assume that we have constructed ζ k satisfying the above properties for J = {1, . . . , k}, we construct ζ k+1 . Consider a point x k+1 ∈ Ω \ ∪ jĺk B 2r j (x j ). Let γ = γ t k+1 be as in Lemma . . We fix λ : [0, ∞) → [0, ∞) such that λ(θ) = 0 if θ ĺ 1 and λ(θ) = 1 if θ ľ 2. Let α be as in ( . ). Let 0 < ε < 1 be be fixed later, and set

ψ ε (x) := γ • λ((ε/|x -x k+1 |) α ), ∀ x ∈ B 2ε (x k+1 ),
extended with the value t k+1 outside B 2ε (x k+1 ). Clearly,

ψ ε is smooth in R N \ {x k+1 }. If we set v ε := π • ψ ε , then v ε -z ∈ C ∞ (R N \ {x k+1 })
and is supported in B ε (x k+1 ).

We claim the following:

ψ ε ∈ W s,p (B 2ε (x k+1 )), ∀ ε > 0, ( . ) ||v ε -z|| W s,p → 0 as ε → 0.

( . )

Taking temporarily for granted the two above properties, we conclude as follows. By ( . ) and Lemma . , for su ficiently small ε we have ||u k + v ε -z|| W s,p ĺ ||u k || W s,p + 2 -k . We complete the proof via a straightforward inductive process, by setting u k+1 := u k + v ε -z.

Proof of ( . ). Assume, for simplicity, that x k+1 = 0. If |x|ĺ 2 -1/α ε, then λ((ε/|x|) α ) = 1. Therefore, with r := 2 -1/α ε, we have where we have used successively Lemma . , ( . ), and Lemma . . Proof of ( . ). By construction, γ is 1-Lipschitz. We find that |∇ψ ε (x)|À ε α /|x| α+1 , and therefore ||∇ψ ε || sp → 0 as ε → 0. Therefore, π • ψ ε -z → 0 in W 1,sp (R N ) as ε → 0. Since the ψ ε 's are uniformly bounded, we conclude via the Gagliardo-Nirenberg inequality ( . ). QED

QED

  Lemma . . Let u ∈ VMO(Ω; F ), where F ⊂ R . Let ρ ∈ C c (B 1 (0); R + ) be such that ´ρ = 1. Then lim ε→0 sup x∈Ωε d(u * ρ ε (x), F ) = 0.

  , a continuous li ting ζ (given by ζ(θ) := ϕ(b j + εe ıθ )) such that ζ(-b) = ζ(b), a contradiction.

  (a) what is a topological singularity? (b) can one detect such singularities? (c) do standard properties of Sobolev spaces hold for maps in W s,p (Ω; N ) without topological singularities? In full generality, the answer to question c) is negative (seeBethuel and Demengel [ ],Bethuel [ ], or [ ] for examples of smooth maps with no extensions or li tings). Depending on the answer we choose to question a), the answer to question (b) could be positive. However, a full theory allowing to encode singularities and/or to clarify their role as only possible obstructions is, for the time being, out of reach. Let us mention several topological invariants that have been investigated so far in the literature: (i) Brouwer degree of maps f : S k → S k (starting with Brezis, Coron, and Lieb [ ]); (ii) spherical homology (starting with Giaquinta, Modica, and Souček [ ]); (iii) (Hopf) degree of maps f : S 3 → S 2 (starting with Rivière [ ]); (iv) higher homotopy groups of general manifolds, under restrictive assumptions on the lower homotopy groups (Pakzad and Rivière [ ]); (v) rational homotopies(Hardt and Rivière [ ]).

.

  The distributional Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The range of the distributional Jacobian . . . . . . . . . . . . . . . . . . . . . . . Inserting singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of the closure of smooth maps . . . . . . . . . . . . . . . . . . . Overview of the higher co-dimensional case . . . . . . . . . . . . . . . . . . . .

  deg(u, a) = N Ju(ϕ), so that ( . ) holds.

  j ), (N j )) ľ max j (ϕ(P j ) -ϕ(N j )); ϕ ∈ Lip 0 (Ω), |ϕ| Lip ĺ 1 . ( . )Remarkably, we actually have equality in ( . ) (see [ ] for the original result and, for other proofs, Brezis [ ] and [ ]): L((P j ), (N j )) = max j (ϕ(P j ) -ϕ(N j )); ϕ ∈ Lip 0 (Ω), |ϕ| Lip ĺ 1 .

  Recall that we consider maps u : Ω ⊂ R N → S N -1 , with N ľ 2. The main result we present here is due to Demengel [ ] when s = 1, 1 ĺ p < 2, and N = 2, Bethuel [ ] when s = 1 and p = N -1, and has been announced, with indications of proof, byBethuel, Coron, Demengel, and Coron [ ] when s = 1 and N -1 < p < N , respectively Mucci [ ] when 0 < s < 1. See alsoPonce and Van Scha tingen [ ].

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The distributional Jacobian. Disintegration (slicing) . . . . . . . . . . . . . . . . . . . The range of the distributional Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of the closure of smooth maps . . . . . . . . . . . . . . . . . . .

  , Runst and Sickel [ ], Maz'ya [ ], Leoni [ ]; an elementary, but partial, account can be find in [ ]. See also the specific references indicated below.

  , see, for example, [ , Appendix]. In the limiting case sp = N , we have the following substitute of the Sobolev non-embedding W s,p → L ∞ (Brezis and Nirenberg [ ], with roots in Boutet de Monvel and Gabber [ ]). Theorem . . Assume that sp ľ N . Then W s,p → VMO, i. e., u ∈ W s,p =⇒ lim ε→0 sup x∈Ωε Bε(x) Bε(x) |u(y) -u(z)| dydz = 0. ( . ) For a proof, see, e. g., [ , Proof of Lemma . ].

  Theorem . . (Inverse trace theory in fractional Sobolev spaces) Let s > 0 be non-integer. We have, with C = C(s, p, N ),|U | W s+1/p,p ĺ C|u| W s,p , ∀ u : R N → R.( . )Theorem . . (Direct trace theory in fractional Sobolev spaces) Let s > 0 be non-integer.

QED

  Corollary . . Let s, p be such that sp ľ dim M . If u j , u ∈ C ∞ (M ; N ) and u j → u in W s,p then, for large j, u j and u are homotopic in C(M ; N ). Proof. It su fices to combine Lemma . with the continuous embeddings W s,p → VMO (Theorem . ) and W s,p → L 1 . QED Appendix # . Standard examples of maps in Sobolev spaces Contents . Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power-type functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gluing maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dipole construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Let Ω := B 1 (0) ⊂ R N . Let α > 0 and set ϕ(x) := 1 |x| α , ∀ x ∈ Ω \ {0}. Then ϕ ∈ W s,p (Ω) ⇐⇒ (α + s)p < N.

I ĺ 2

 2 ˆ1 0 t N -(α+s)p-1 dtˆRN|δ M e ϕ(y)| p dy.Note that the assumption on α implies that ´1 0 t N -(α+s)p-1 dt < ∞. On the other hand, we have ´RN |δ M e ϕ(y)| p dy < ∞, since |δ M e ϕ(y)|∼ 1 |y| α+M at infinity and |δ M e ϕ| p has integrable singularities. (These singularities, located at y = 0, -e, ..., -M e, behave like |x| -αp .) This proves " ⇐= "." =⇒ " We note that ( . ) implies that I ľ 2 ˆ1 0 ˆB1 (0) t N -(α+s)p-1 |δ M e ϕ(y)| p dydt = ∞.

  u(x) := x |x| , ∀ x ∈ R 2 \ {0}. Let s > 0, 1 ĺ p < ∞. Then, u ∈ W s,p (B 1 (0)) if and only if sp < 2.b) More generally, let k ∈ {1, . . . , N } and letu : R N → R k , u(x 1 , . . . , x N ) := (x 1 , . . . , x k ) |(x 1 , . . . , x k )| , ∀ x ∈ R N \ ({0} × R N -k ).

Lemma . .

 . Let U be a neighborhood of Ω and let S be a d-dimensional submanifold of U , withd ĺ N -1. Let u ∈ C ∞ (U \ S) satisfy |D u(x)|ĺ C [dist (x, S)] -, = 0, . . . , N -d, ∀ x ∈ U \ S.( . )Then, u |Ω ∈ W s,p (Ω) provided sp < N -d.

Lemma . .

 . Let S be a d-submanifold of the open set Ω ⊂ R N , with d ĺ N -2. Let u ∈ C 1 (Ω \ S) be such that ∇u ∈ L 1 loc (Ω). Then u ∈ W 1,1 loc (Ω).Proof of Lemma . continued. Combining ( . ) with ( . ) and Lemma . , we find that D j u ∈ L p (Ω), j = 0, . . . , m. It remains to check thatˆΩ ˆΩ |D m u(x) -D m u(y)| p |x -y| N +σp dxdy < ∞. ( . )For this purpose, we note that, with constants depending on u but not on x or y, we have:|D m u(x) -D m u(y)|ĺ C max |x m u(x) -D m u(y)| p |x -y| N +σpdx dy = 2(I + J), where I := ¨|x-y|<d(y)ĺd(x) . . . , J := ¨d(y)ĺmin{d(x), |x-y|} . . . Using ( . ) (respectively ( . )), in order to estimate I (respectively J), we find that ¨|D m u(x) -D m u(y)| p |x -y| N +σp dx dy À ¨|x-y|<d(y)ĺd(x) |x -y| p-N -σp d (m+1)p (y) dx dy + ¨d(y)ĺmin{d(x), |x-y|} dx dy |x -y| N +σp d mp (y) À ˆ1 d mp+σp (y) dy < ∞, by ( . ) (since mp + σp < N -d). QED Remark . . In Lemma . , when d ĺ N -2, the assumptions on u can be weakened to u ∈ C N -d (Ω \ S) and |D N -d u(x)|ĺ C[dist (x, S)] d-N .

  dH N -1 (x) |x -(1 + t)e N | β .

  |ψ ε | p W s,p ľ ˆBr(0) ˆBr(0) d E (γ((ε/|x|) α ), γ((ε/|y|) α )) p |x -y| N +sp dxdy = ˆBr(0) ˆBr(0) |(ε/|x|) α -(ε/|y|) α )| p |x -y| N +sp dxdy = ∞,

.

  Slicing and characterization via di ferences . . . . . . . . . . . . . . . . . . . . . Sobolev embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gagliardo-Nirenberg inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . Trace theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superposition operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative suboptimal Sobolev embeddings . . . . . . . . . . . . . . . . . . . . Martingales and Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adapted trace theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homotopy and VMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	. .

Motivation. Program. Preliminary remarks

We discuss a few natural questions concerning the Sobolev spaces W s,p (Ω; N ), in the following setting: a) N is a smooth connected closed Riemannian manifold isometrically embedded into some R . b) 0 < s < ∞, 1 ĺ p < ∞. c) Ω ⊂ R N is "smooth" and bounded. In most cases, Ω is a ball or a cube.

. .

The dipole construction

We describe and analyze here the dipole construction of Brezis, Coron, and Lieb [ ], in a form and functional setting adapted to our purposes. (ii) a ∈ S N -1 .

(iii) f ∈ C ∞ ([0, 1]; [0, 1]) be such that f (0) = f (1) = 0, f (0) > 0, f (1) < 0, and f (θ) > 0, ∀ θ ∈ (0, 1).

(iv) v ∈ C ∞ (R N -1 ; S N -1 ) be such that v(x) = a if |x|ľ 1/2 and deg v = 1.

Set, for 0 < ε ĺ 1,

.

Then

)

supp(u ε -a) ⊂ {(x , θ) ∈ R N ; 0 ĺ θ ĺ 1, |x |ĺ εf (θ)}, ∀ ε.

( . )

Proof when N ľ 3. We present the proof when N ľ 3. As we will see, it relies on a Gagliardo-Nirenberg embedding that fails when N = 2. For a full proof (including the case N = 2), see [ , ].

Step . Preliminary remarks. Property ( . ) is clear. It is also clear that u ε is smooth except at 0 and e N . Moreover, we claim that deg(u ε , 0) = 1 (and, similarly, deg(u ε , e N ) = -1). Indeed, let δ > 0 be small. By a homotopy argument,

and the latter number is 1, by the definitions of v and u ε .

The claim, combined with ( . ) and the fact that, as we will see, u ε ∈ R 0 , implies ( . ).

Step . The main estimate and conclusion. Set

By a tedious calculation, using the fact that |x /(εf (θ))|ĺ 1 when x ∈ V ε , we find that

On the one hand, using ( . ) and the assumptions on f , we find that u ε ∈ R 0 . On the other hand, ( . ) implies that, for

In particular, ( . ) holds when s = N -1. By the Gagliardo-Nirenberg inequality (Corollary . ) and the fact that N ľ 3, we find that ( . ) still holds in the full range given in ( . ).

QED

By scaling, Lemma . implies the following result.

Lemma . . (Dipole construction, scaled version) Let N , a, f , v, s, p be as above. Let A, B ∈ R N . Set ξ := (B -A)/|B -A| and H := ξ ⊥ , oriented by an orthonormal basis (e 1 , . . . , e N -1 ) such that (e 1 , . . . , e N -1 , ξ) is a direct basis of R N . Let R be an orientation preserving linear isometry from H to R N -1 . Write a point x in R N as x = x + θξ, with x ∈ H and θ ∈ R. Let L := |B -A| and set, for 0 < ε ĺ L,

.

Then

)

We next explain how to "make room" for inserting a dipole into an already existent map u : Ω → S N -1 . Lemma . . (Making room when u is locally smooth) Let N ľ 3. Let 0 < s < N -1 and 1 < p < ∞ be such that sp = N -1. Let a ∈ S N -1 . Let γ be a smooth simple compact curve in R N . Let u : R N → S N -1 be smooth in an open neighborhood U of γ. Fix δ, µ > 0. Then, for small ε > 0, there exists a map r u : R N → S N -1 such that:

Similarly in the neighborhood of a finite union of smooth simple compact curves.

Proof.

Step . Making u constant near its endpoints. Assume, for simplicity, that the origin is an end-

, it is easy to see (using the assumption that u is smooth) that, for small ε, r u satisfies a)-c) with γ replaced by 0, and, in addition, D N -1 (r u -u) 1 → 0 as ε → 0. By the Gagliardo-Nirenberg inequality Corollary . (recall that N ľ 3 and that u is bounded), we find that d) holds as well.

Therefore, in what follows, we may assume that u is constant near the endpoints of γ.

Step . Construction of r u and conclusion. Since N ľ 3, the set u(γ) ∪ {a} is contained in the interior of a closed spherical cap Σ = S N -1 . By a standard argument, there exists a smooth map

Set, for small ε,

Clearly: (j) the definition is consistent when

then conclude via Gagliardo-Nirenberg. In order to prove ( . )-( . ), we note that: (i) thanks to the assumption that u is constant near the endpoints of A and B of γ, we may replace V ε with the smaller set

Combining these facts with the definition ( . ), we find that

Integrating ( . ), we find, with the help of the coarea formula, that

Let γ be a smooth simple compact curve in R N . Assume that 0 is one of the endpoints of γ, and that γ (0) = e N .

Set, for α, β > 0,

Assume that:

(i) There exist α > 0, β > 0, and a ∈ S N -1 such that

Fix δ, µ > 0. Then, for small ε > 0, there exists a map r u : R N → S N -1 such that:

Sketch of proof. The proof is essentially the same as the one of Lemma . , with, as additional ingredient, the use of property (iii) of the homotopy H. The definition ( . ) has to be modified to

. Details are le t to the reader.

QED

Lemma . . (Making room near a singular endpoint ( )) Let:

(iii) µ > 0.

(iv) u : R N → S N -1 .

Assume that, in the unit ball B, u belongs to R 0 . Let 0 < δ < 1 be such that u is smooth in B δ (0) \ {0}. Then there exists some map r u : R N → S N -1 such that:

e) ||r u -u|| p W s,p ĺ µ.

Similarly for N -valued maps, provided we replace conclusion c) with "r u and u are homotopic on small spheres around the origin".

Proof. We write points in R N \ {0} in the form x = rσ, r > 0, σ ∈ S N -1 . Consider a map w ∈ C ∞ (S N -1 ; S N -1 ) of degree deg(u, 0) and such that w = a near the North Pole e N . Let H ∈ C ∞ (S N -1 × (-∞, δ]; S N -1 ) be such that:

Let, for small ε,

Note that: (j) the definition is consistent when εδ/2 ĺ |x|ĺ εδ; (jj) a)-d) hold. Next, we note that

Integrating ( . ), we find that ||r u -u|| W N -1,1 → 0 as ε → 0. We conclude via the Gagliardo-Nirenberg inequality.

Assume that, for each cube C ∈ C N , g |∂C : ∂C → N is null homotopic. Let f : C N → N be the (N -1)-homogenous extension of g. Then there exists a sequence

Sketch of proof. We work with |x|:= ||x|| ∞ . Let δ be the size of the cubes in C N . Fix some point a ∈ N . Consider, for each cube

The fact that, for fixed arbitrarily small µ > 0, may choose the ε j 's such that r f -f W s,p ĺ µ follows from the multi-sequences Brezis-Lieb Lemma . .

QED

Appendix # . More on homogeneous maps

For the next result, we use the notation in Section . . Let Ψ = Ψ j,t,ε : R N → C j = C j,t,ε be the projection on the j-skeleton of size 2ε of R N obtained from C N,t,ε . Set v := t + (ε, . . . , ε).

Lemma . . The mapping Ψ : R N \ C N -j-1,v,ε → C j,t,ε is locally Lipschitz, and satisfies, with C independent of ε and t,

Sketch of proof. By scaling, we may assume that ε = 1 and t = 0. In this case, C := C N -j-1,v,1 is given by

For further use, let us also note that, if

and τ m x m ľ 0, ∀ 1 ĺ m ĺ N -j}. . ) and a similar formula holds for a general σ ∈ S N -j,N .

Using ( . ) and its analogues, one can easily prove that Ψ is continuous on R N \ C . On the other hand, the sets S σ,τ form a polyhedral a.e.-partition of Q 1 and, in the interior of S id,τ and for k ∈ Z N we have (using ( . ) and ( . ))

(and similar formulas hold in each S σ,τ ). Combining ( . ) and its analogues, we obtain ( . ). QED For the next result, we are in the context of Proposition . , and we use the notation there.

Lemma . . Let 1 ĺ p < j + 1 ĺ N . Assume that g ∈ W 1,p (C j ; R ). Then we have

)

Equivalently, the map W 1,p (C j ) g → g j ∈ W 1,p (C ) is continuous.

Proof. By a straightforward induction argument, it su fices to prove that (with the notation at the beginning of the Section . ) the map h := H j+1 (g) satisfies

)

The validity of ( . ) on each cube C ∈ C j+1 , and thus on C j+1 , is clear (here, we use p < j + 1). We next check ( . ) first "cube by cube", next globally. Fix first a cube

. (Here, we use again the fact that p < j + 1.) The case of a general map g ∈ W 1,p (∂C) follows by approximation.

It remains to prove that h ∈ W 1,p (C j+1 ) (globally). This amounts to tr(h |C ) = g |∂C , ∀ C ∈ C j+1 . This is clear when g |∂C is Lipschitz; the general case follows by approximation. Lemma . . Assume that E is non-compact. Then, for each t ∈ E , there exists some smooth map γ

For the next result, we are in the context of Propositions . and . , and we use the notation there.

Lemma . . Let 0 ĺ j ĺ N -1. Then there exists a Lipschitz homotopy G = G(x, θ) :

Proof. The proof is by induction on N -j and "concatenation" of successive homotopies. When

Assume next that N -j = k ľ 2 and the the lemma has been established for N -1 and j. Let us note that

Using this, we see that

is a homotopy satisfying conclusions a) and c) above. Moreover, if we set r

It then su fices to concatenate G 1 with an appropriate map (given by the induction hypothesis)

In what follows, we let Ω = (0, 1) N .

For the next result, see [ , Theorem B. ].

Lemma . . Let F ⊂ R k be a discrete closed set. Let f ∈ W s,p (Ω; F ). If sp ľ 1, then f is constant.

Lemma . . Assume that sp ľ 1. Let ϕ ∈ W s,p (Ω; E ) be such that u := π • ϕ is continuous. Then ϕ is continuous. Moreover, if u is smooth, then ϕ is smooth.

Proof. We may assume that s ĺ 1. Given the local nature of the problem, we may also assume that u(Ω) ⊂ B for some geodesic ball or radius r smaller than the injectivity radius of N . Write π -1 (B) as an at most countable union of balls B j , each one di feomorphic with B. Pick j such that the set ϕ -1 (B j ) has positive measure. Let ζ be the continuous li ting of u with values into B j . By Theorem . , we have ζ ∈ W s,p and, by choice of B j , the set [ϕ = ζ] has positive measure. We claim that ϕ = ζ a.e. (and this completes the proof).

For this purpose, we consider some smooth map g : [0, ∞) → [0, 1] such that g(0) = 0 and g(θ) = 1 when θ ľ r. Let ( . )

Since ϕ, ζ ∈ W s,p and g is Lipschitz, we obtain from ( . ) that f ∈ W s,p (recall that s ĺ 1).

On the other hand, we clearly have, by the formula of g and the property π • ϕ = π • ζ, that f : Ω → {0, 1}. By Lemma . and the choice of B j , we find that f = 0 a.e., and thus ϕ is continuous. QED