Haïm Brezis

A brief history of the Jacobian

Keywords: distributional Jacobian, divergence structure MSC 2020: 01-01, 01A55, 01A60, 42B35

In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector fields involving the cofactors of the Jacobian are divergence free. We illustrate the fundamental impact of these properties on research, from the times of Jacobi to our days.

Introduction

Given a mapping u : Ω ⊂ R N → R N of class C 1 , recall that its Jacobian is given by det (∇u) = N j=1 (∂ j u i ) C i, j , i ∈ {1, . . . , N}, (

where C i, j denotes the cofactor of ∂ j u i in the matrix (∂ u k ) ,k .

This short expository text is a tribute to two remarkable (families of) identities related to the Jacobian of u, with u of class C 2 , which appear frequently in the literature, usually without credit. The first one asserts that det (∇u) = N j=1 ∂ j (u i C i, j ), i ∈ {1, . . . , N}, (

and in particular implies that the Jacobian has a divergence structure. The second one is N j=1 ∂ j C i, j = 0, i ∈ {1, . . . , N}. (0.3) These (equivalent) identities were obtained by Jacobi himself and rediscovered, under possibly different but equivalent forms, a number of times; see Section 1 below.

The distributional Jacobian -which grew out of the identity (0.2) -plays an important role in topics from the calculus of variations, arising, e.g., in nonlinear elasticity, and developed in the seminal works of Morrey [START_REF] Morrey | Multiple Integrals in the Calculus of Variations[END_REF], Reshetnyak [START_REF] Yu | Mappings with bounded distortion as extremals of integrals of Dirichlet type[END_REF], and Ball [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF]. In this context, the proofs use the continuity of the mapping u → det (∇u), property which relies heavily on the divergence structure of the Jacobian. Section 2 contains some optimal versions of the continuity properties of the (distributional) Jacobian. In Sections 3 and 4, we present a partial collection (reflecting our personal taste!) of topics, motivated, e.g., by liquid crystals, Ginzburg-Landau theories, surfaces of constant mean curvature, which involve (0.2) and (0.3) and illustrate their important influence on current research.

The origins of a formula

Carl Gustav Jacobi (1804-1851) is famous for his pioneering work on elliptic functions, but most students in mathematics learn his name through the concept of Jacobian of a C 1 mapping of R N into R N , which is the determinant of the matrix of the first order partial derivatives of its components. The term 'Jacobian' refers to an important memoir (in Latin) of Jacobi on functional determinants (his terminology) published in 1841 [START_REF] Jacobi | De determinantibus functionalibus[END_REF]. Jacobians occur in an essential way in fundamental questions of analysis like the implicit functions theorem and the change of variables in multiple integrals. The name 'Jacobian' was coined in 1853 by James Joseph Sylvester (1814-1897) [48, p. 476]: 'where J indicates the Jacobian of the given functions f and g in respect to the variables x and l, meaning thereby the so-called Functional Determinant of Jacobi to f and g in respect of x and l'. The memoir [START_REF] Jacobi | De determinantibus functionalibus[END_REF] of Jacobi provided the first systematic treatment of the principal properties of the Jacobians in arbitrary finite dimension.

A few years later, in 1844-45, Jacobi published a substantial memoir [START_REF] Jacobi | Theoria novi multiplicatoris systemati aequationum differentialum vulgarium applicandi, Pars I[END_REF] on the method of the multiplier, which extends to systems of ordinary differential equations the method of the integrating factor for scalar equations. Consider an autonomous system x = X (x) of N ordinary differential equations, i.e.,

x i = X i (x 1 , . . . , x N ), i ∈ {1, . . . , N}, (1.1)
and (N -1) functionally independent first integrals of (1.1),

θ i = θ i (x 1 , . . . , x N ), i ∈ {1, . . . , N -1}.
Let θ = θ(x 1 , . . . , x N ) be a C 1 function, and write

det (∇θ, ∇θ 1 , . . . , ∇θ N-1 ) = N j=1 (∂ j θ) ∆ j .
Clearly, the cofactor

∆ j = ∆ j (x 1 , . . . , x N ) of ∂ j θ depends only on θ 1 , . . . , θ N-1 . Note that N j=1 (∂ j θ i ) ∆ j = 0, i ∈ {1, . . . , N -1}, (1.2) and, since each θ i is a first integral, N j=1 (∂ j θ i ) X j = 0, i ∈ {1, . . . , N -1}. (1.3)
Assuming that X (x) = 0, we find, from (1.2), (1.3), and the functional independence of θ 1 , . . . , θ N-1 , that there exists some M(x) = 0 such that M(x)X j = ∆ j , j ∈ {1, . . . , N}. Granted (0.3), one obtains the differential equation satisfied by M, N j=1 ∂ j (M X j ) = 0.

(1.4)

The paramount importance of M (the celebrated Jacobi multiplier) comes from the fact that, if M, M are solutions of (1.4), the quotient M/M is a first integral of (1.1).

The validity of (0.3) was stated and proved by Jacobi on p. 203 of the first part of [START_REF] Jacobi | Theoria novi multiplicatoris systemati aequationum differentialum vulgarium applicandi, Pars I[END_REF]. With the notation used in the introduction, the result, that Jacobi named 'fundamental lemma', is the following.

Lemma 1.1. If u : Ω ⊂ R N → R N is of class C 2 , then N j=1 ∂ j C i, j = 0, i ∈ {1, . . . , N}.
Jacobi's proof relies on differentiation of functional determinants defined classically as alternate sums of products of the first partial derivatives of the u i . At the beginning of his proof [30, p. 203], Jacobi observed that the formula (0.2) can be deduced from the formula (0.3). Indeed, the elementary identity

N j=1 ∂ j (u i C i, j ) = N j=1 (∂ j u i )C i, j + u i N j=1 ∂ j C i, j = det (∇u) + u i N j=1 ∂ j C i, j , i ∈ {1, . . . , N},
combined with (0.3) yields (0.2).

On the other hand, note that (by construction) C i, j is independent of u i , ∀ i, ∀ j. Therefore, if we define u by u j := u j for j = i and u i := 1 we have (with obvious notation) C i, j = C i, j , ∀ j. Applying (0.2) to u yields

0 = det (∇ u) = N j=1 ∂ j C i, j = N j=1 ∂ j C i, j , so that (0.3) holds.
An excellent analysis of Lemma 1.1 and of Jacobi's proof is proposed by T. Muir on p. 230-235 of the second volume of his monumental treatise on the history of determinants [START_REF] Muir | The Theory of Determinants in the Historical Order of Development[END_REF].

In his lectures on dynamics delivered in 1842-43, posthumously published in 1866, reproduced (slightly revised) in 1884 as a supplementary volume of his Gesammelte Werke [START_REF] Jacobi | Vorlesungen über Dynamik. Gehalten an der Universität zu Königsberg im Wintersemester 1842-1843[END_REF], and translated in English in [START_REF] Jacobi | Jacobi's Lectures on Dynamics[END_REF], Jacobi developed in Lecture 13 the theory of functional determinants, essentially following [START_REF] Jacobi | De determinantibus functionalibus[END_REF], and, in Chapter 14, the theory of the multiplier. At this occasion, in [31, p. 104-106] (or [32, p. 112-114]), he proved again Lemma 1.1, without isolating it as a separate statement.

The multiplier rapidly became a classical tool in analysis and analytical mechanics and, in particular, it inspired to Henri Poincaré (1854Poincaré ( -1912) ) his theory of the integral invariants [START_REF] Poincaré | Les méthodes nouvelles de la mécanique céleste, tome III. Invariants intégraux. Solutions périodiques de deuxième espèce[END_REF]. The Section V of Chapter XXV of the monumental treatise of rational mechanics of Paul Appell (1855Appell ( -1930) ) [START_REF] Appell | Traité de mécanique rationnelle, tome deuxième. Dynamique des systèmes. Mécanique analytique[END_REF] is devoted to the Jacobi multiplier. Lemma 1.1 is proved in [4, p. 460-461] following the argument of Jacobi sketched above. Lemma 1.1 has also found applications outside of the fields of ordinary differential equations and analytical mechanics. In 1910, Jacques Hadamard (1865Hadamard ( -1963) ) extended, in an Appendix to Tannery's textbook on real functions [START_REF] Hadamard | Sur quelques applications de l'indice de Kronecker[END_REF], the notion of Kronecker index to continuous non vanishing mappings u on the smooth boundary ∂Ω of a bounded open set Ω ⊂ R N . At this occasion, he stated and used in [25, p. 455-456] the following generalization of the formulas (0.2) and (0.3)), that we formulate keeping the notation and assumptions of Lemma 1.1.

Lemma 1.2. For any h

= h(y) = h(y 1 , . . . , y N ) : R N → R N of class C 1 , one has N j=1 ∂ j N k=1 (h k • u) C k, j = N k=1 (∂ k h k ) • u • det (∇u), (1.5)
where, on the left-hand side, ∂ j = ∂/∂x j , while, on the right-hand side,

∂ k = ∂/∂ y k .
Notice that if, in (1.5), we take h k (y) := δ k j y j , k, j ∈ {1, . . . , N}, we recover the formulas (0.2), while, if we take h k (y) := δ k j , k, j ∈ {1, . . . , N}, we recover the formulas (0.3).

Hadamard's 'proof' of this 'easy to verify' identity consists in a footnote asserting that the formula is 'well-known when the h k 's are constant and plays a role in the theory of multipliers' (referring to the classical treatises on analysis by Jordan, Goursat, etc.). It is doubtful that the majority of readers of Tannery's textbook containing Hadamard's paper [START_REF] Hadamard | Sur quelques applications de l'indice de Kronecker[END_REF] were able to reconstruct a proof of (1.5) from this rough information. We sketch an argument at the end of this section.

The formulas (0.3) reappear almost fifty years later in the analytical definition of the Brouwer degree (an extension of Kronecker's index; see, e.g., Dinca and Mawhin [START_REF] Dinca | The Core of Nonlinear Analysis[END_REF]) given by Erhard Heinz (1924Heinz ( -2017) ) in [27, p. 232]. They are just referred there as 'wellknown relations' and stated without proof. They are essential to justify Heinz' definition of the Brouwer degree of u in Ω as some integral over Ω of an expression depending on u and det (∇u).

As noticed in Brezis and Nguyen [START_REF] Brezis | The Jacobian determinant revisited[END_REF], one also finds the formulas (0.3) (or rather the version (1.6) below) in the monograph of Charles B. Morrey (1907Morrey ( -1984) ) on the calculus of variations and partial differential equations [START_REF] Morrey | Multiple Integrals in the Calculus of Variations[END_REF]Lemma 4.4.6,p. 122], which has led the authors of [START_REF] Brezis | The Jacobian determinant revisited[END_REF] to attribute them to Morrey. The proof in [START_REF] Morrey | Multiple Integrals in the Calculus of Variations[END_REF] is by induction on N, and the motivation of Morrey is the study of strongly quasi-convex functions.

Another identity involving Jacobians is fundamental in the analytical proof by Dunford and Schwartz [23, p. 467] of the Brouwer fixed point theorem [23, p. 468-470].

Lemma 1.3. If f : Ω → R N-1 is of class C 2 and
, for each j ∈ {1, . . . , N}, E j denotes the determinant whose columns are the (N -1) partial derivatives

∂ 1 f , . . . , ∂ j f , . . . , ∂ N f , then N j=1 (-1) j-1 ∂ j E j = 0.
(1.6) (As usual, ∂ j f indicates that the factor ∂ j f is missing.)

Proof. It turns out that (1.6) is actually equivalent to (0.3). Indeed, if u is as in Lemma 1.1 and we set f := (u 1 , . . . , u i , . . . , u N ) :

Ω → R N-1
, then E j = (-1) i+ j C i, j , j ∈ {1, . . . , N}, and thus (1.6) implies (0.3). On the other hand, if f is as in Lemma 1.3 and we let u : Ω → R N be any C 2 mapping whose first (N -1) components are those of f , then C N, j = (-1) N+ j E j , j ∈ {1, . . . , N}, and therefore (0.3) implies (1.6).

The modern 'textbook' analytical proofs of the Brouwer fixed point theorem (see, e.g., Kannai [START_REF] Kannai | An elementary proof of the no-retraction theorem[END_REF] or Evans [START_REF] Evans | Partial differential equations[END_REF]Section 8.1.4]) follow the lines of [START_REF] Dunford | Linear Operators. Part I General Theory[END_REF] and rely on the formulas (0.3) or cousin formulas, as (1.6).

Finally, let us mention -as suggested to us by Ball -that the formulas (0.3) establish a connection between Jacobians and null Lagrangians. More specifically, they imply that L(P) := det P, where P is an N × N matrix, is a null Lagrangian. (More generally, (0.3) implies that L(P, z) := η(z) det P, with η : R N → R of class C 1 , is a null Lagrangian.) This is well explained in [START_REF] Evans | Partial differential equations[END_REF]Section 8.1.4], where the reader may find a proof of (0.3). For more insights on the structure of null Lagrangians, see, e.g., Olver and Sivaloganathan [START_REF] Olver | The structure of null Lagrangians[END_REF] and the survey by Iwaniec [START_REF] Iwaniec | Null Lagrangians, the Art of Integration by Parts[END_REF].

This confirms the ubiquity of (0.3). Some final comments. The somewhat mysterious character of the formulas (0.2) (or, equivalently, (0.3)) disappears when they are written in the language of exterior differential calculus, where they take the form of the trivial results

d (-1) i-1 u i du 1 ∧ . . . ∧ du i ∧ . . . ∧ du N = du 1 ∧ . . . ∧ du N , i ∈ {1, . . . , N},
where d denotes the exterior differential, ∧ the exterior product, and du i means that the factor du i is missing. On the other hand, the identities (0.3) correspond to the trivial property d(du 1 ∧ . . . ∧ du i ∧ . . . ∧ du N ) = 0, i ∈ {1, . . . , N}, which follows from the fact that d 2 u j = 0, ∀ j. This is the reason why the formulas (0.2) and (0.3) do not appear explicitly in the analytical presentation of the Kronecker index and of the Brouwer degree in terms of differential forms, given for example in [START_REF] Dinca | The Core of Nonlinear Analysis[END_REF].

Here are some more detailed explanations. It is well-known that

du 1 ∧ . . . ∧ du N = det (∇u) dx 1 ∧ . . . ∧ dx N . (1.7)
On the other hand, one may check that

du 1 ∧ . . . ∧ du i ∧ . . . ∧ du N = N j=1 (-1) i+ j C i, j dx 1 ∧ . . . ∧ dx j ∧ . . . ∧ dx N . (1.8)
Using (1.7), (1.8), the exterior differentiation rules, and the fact that d 2 u j = 0, ∀ j, we find that

det (∇u) dx 1 ∧ . . . ∧ dx N =du 1 ∧ . . . ∧ du N =d[(-1) i-1 u i du 1 ∧ . . . ∧ du i ∧ . . . ∧ du N ] =d N j=1 (-1) j-1 u i C i, j dx 1 ∧ . . . dx j ∧ . . . ∧ dx N = N j=1 ∂ j (u i C i, j ) dx 1 ∧ . . . ∧ dx N , whence (0.2).
Similarly, the formula (1.5) of Hadamard amounts to the straightforward identity

d N k=1 (-1) k-1 h k • u du 1 ∧ . . . ∧ du k ∧ . . . ∧ du N = N k=1 (∂ k h k ) • u du 1 ∧ . . . ∧ du N .

The distributional Jacobian

For simplicity, we assume throughout the rest of this paper that Ω is smooth and bounded.

Taking advantage of the divergence structure (0.2) of the Jacobian, Ball [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF] introduced in 1976 the very useful concept of Jacobian in the sense of distributions, nowadays rather known as distributional Jacobian. [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF] deals only with the cases N = 2 and N = 3 (see (6.8) and (6.10) in [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF]), but the definitions readily extend to any N ≥ 2. More precisely, given a vector-field u :

Ω → R N , set Det (∇u) = T 1 := N j=1 ∂ j (u 1 C 1, j ) (2.1)
whenever the right-hand side of (2.1) is meaningful as a distribution, and, in particular, if u 1 C 1, j ∈ L 1 (Ω), ∀ j. This is, e.g., the case if:

u ∈ W 1,N-1 (Ω; R N ) ∩ L ∞ (Ω; R N ) (2.2) or u ∈ W 1,N 2 /(N+1) (Ω; R N ) (2.3) (thanks to the Sobolev embedding W 1,N 2 /(N+1) (Ω) → L N 2 (Ω)).
As noted in [5, Corollary 6.2.1], if u satisfies (2.2) or (2.3), one has

Det (∇u) = T i := N j=1 ∂ j (u i C i, j ), i ∈ {2, . . . , N}. (2.4)
The identity (2.4) easily follows from (0.2) and a standard approximation argument.

Another sufficient condition for the validity of (2.4) is |u| |∇u| N-1 ∈ L 1 (Ω) -which is weaker than (2.2) and (2.3). This assertion is a consequence of a more general result due to Mironescu [START_REF] Mironescu | A remark on the distributional Jacobian[END_REF].

In recent years, it has become customary to use a more symmetric definition of the distributional Jacobian:

Det (∇u) = 1 N N i=1 T i = 1 N N i=1 N j=1 ∂ j (u i C i, j ), (2.5)
or, equivalently,

Det (∇u) = 1 N div D, (2.6)
where D = D(u) = (D 1 , . . . , D N ) and

D i = D i (u) := det (∂ 1 u, . . . , ∂ i-1 u, u, ∂ i+1 u, . . . , ∂ N u) , i ∈ {1, . . . , N}. (2.7)
Note that, at least when u satisfies (2.2) or (2.3), the formulas (2.1) and (2.6) yield the same distributional Jacobian.

Clearly, (0.2) implies that, when u ∈ C 2 (Ω; R N ), we have Det (∇u) = det (∇u).

(2.8)

One can investigate the validity of (2.8) under weaker assumption on u, assuming at least that u ∈ W 1,1 loc (Ω; R N ) (so that the right-hand side of (2.8) is a function defined a.e.). An approximation argument shows that (2.8) still holds when u ∈ C 1 (Ω; R N ), and thus the distributional Jacobian Det (∇u) generalizes the classical Jacobian det (∇u). Moreover, the equality (2.8) persists when u ∈ W 1,N (Ω; R N ); this follows, again, by approximation. However, Ball noticed that, when

u ∈ W 1,N-1 (Ω; R N ) ∩ L ∞ (Ω; R N ), (2.8) may fail. His example [5, Example 6.1] is the following. Let Ω = B N := {x ∈ R N ; |x| < 1} and u(x) := R(|x|) |x| x, (2.9)
where R is a smooth real-valued function on [0, 1] such that R(0) = 0. Note that u satisfies (2.2). (In fact, u ∈ W 1,p (Ω; R N ), ∀ p < N.) Ball pointed out that 'Det (∇u) has an atom at x = 0', but he didn't elaborate; a more detailed computation yields

Det (∇u) = R N-1 (|x|) R (|x|) |x| N-1 + |B N | R N (0) δ 0 in D (Ω), (2.10)
where |B N | denotes the measure of the unit ball. (We will return to this phenomenon in Section 3 below.)

However, we have the following noticeable result, obtained by Müller [START_REF] Müller | Det = det. A remark on the distributional determinant[END_REF].

Theorem 2.1. Assume that (2.3) holds.

If, in addition, Det (∇u) ∈ L 1 , then Det (∇u) = det (∇u).

More generally, if (2.3) holds and Det (∇u) is a Radon measure, then det (∇u) is (the density of) the absolutely continuous part of Det (∇u) (with respect to the Lebesgue measure).

The distributional Jacobian enjoys remarkable continuity ('stability') properties, which rely heavily on its divergence structure, and are relevant for variational problems (which require passing to the weak limits). Here are some typical examples taken from Brezis and Nguyen [START_REF] Brezis | The Jacobian determinant revisited[END_REF].

Theorem 2.2. Assume that p ≥ N -1 and N -1 p + 1 q = 1. (2.11) For all u, v ∈ W 1,p (Ω; R N ) ∩ L q (Ω; R N ) and ϕ ∈ C 1 c (Ω; R), we have |〈Det (∇u), ϕ〉 -〈Det (∇v), ϕ〉| ≤ C N ||u -v|| L q (||∇u|| L p + ||∇v|| L p ) N-1 ∇ϕ L ∞ , (2.12
)

|〈Det (∇u), ϕ〉 -〈Det (∇v), ϕ〉| ≤C N ||∇u -∇v|| L p (||∇u|| L p + ||∇v|| L p ) N-2 × (||u|| L q + ||v|| L q ) ∇ϕ L ∞ . (2.13)
Estimate (2.12) illuminates the fact that, if u k u weakly in W 1,r (Ω; R N ) for some r > N 2 /(N + 1), then Det (∇u k ) → Det (∇u) in D (Ω).

(2.14) (Indeed, it suffices to apply (2.12) with p = N 2 /(N + 1) and q = N 2 and to use the fact that W 1,r (Ω) → L N 2 (Ω), with compact injection.) This result is due to Reshetnyak [START_REF] Yu | Mappings with bounded distortion as extremals of integrals of Dirichlet type[END_REF] when r = N. His proof works in the full range r > N 2 /(N + 1), and also yields (2.14) in the limiting case r = N 2 /(N + 1), under the extra assumption that u k → u strongly in L N 2 (Ω; R N ). However, it lacks the quantitative estimate (2.12).

Dacorogna and Murat [21, proof of Theorem 1] have shown that, when r = N 2 /(N + 1), the extra assumption that u k → u strongly in L N 2 (Ω; R N ) cannot be omitted; they have constructed a sequence (u k ) ⊂ W 1,N 2 /(N+1) (Ω; R N ) such that u k 0 weakly in W 1,N 2 /(N+1) , while Det (∇u k ) does not converge to 0 in D (Ω) (see also [START_REF] Brezis | The Jacobian determinant revisited[END_REF]Proposition 1] for a more precise conclusion).

It turns out that the space W 1,N 2 /(N+1) is the largest Sobolev space in the scale W 1,p , 1 ≤ p < ∞, for which the distributional Jacobian makes sense. This is a consequence of the following result (see [21, proof of Theorem 1] and [19, proof of Lemma 5, Case 1]).

Theorem 2.3. Given any

1 ≤ p < N 2 /(N +1), there exists a sequence (u k ) ⊂ C ∞ (Ω; R N ) such that, as k → ∞, u k → 0 strongly in W 1,p and Ω det (∇u k ) ϕ → ∞ for some ϕ ∈ C ∞ c (Ω; R).
A more general natural question investigated in Brezis and Nguyen [START_REF] Brezis | The Jacobian determinant revisited[END_REF] is the existence of a largest fractional Sobolev space W s,p in which Det (∇u) is well-defined and stable. The answer is given by the following result [19, Theorem 3, Theorem 4]. Theorem 2.4.

1. If u, v ∈ C ∞ (Ω; R N ) and ϕ ∈ C 1 c (Ω; R), then |〈det (∇u), ϕ〉 -〈det (∇v), ϕ〉| ≤C N,Ω |u -v| W (N-1)/N,N × |u| N-1 W (N-1)/N,N + |v| N-1 W (N-1)/N,N ∇ϕ L ∞ . (2.15)
In particular, there exists a unique continuous mapping

W (N-1)/N,N (Ω; R N ) u → Det (∇u) ∈ D (Ω) such that Det (∇u) = det (∇u) when u ∈ C ∞ (Ω; R N ). 2. Let s > 0 and 1 ≤ p ≤ ∞. If the mapping C ∞ (Ω; R N ) u → det (∇u) ∈ D (Ω)
is continuous with respect to the W s,p -convergence, then W s,p (Ω) → W (N-1)/N,N (Ω).

We recall that, when 0 < s < 1 and 1 ≤ p < ∞, is the Gagliardo seminorm.

The proof of (2.15) relies on the identity (2.17) below, with roots in Bourgain, Brezis, and Mironescu [START_REF] Bourgain | On the structure of the Sobolev space H 1/2 with values into the circle[END_REF] and Hang and Lin [START_REF] Hang | A remark on the Jacobians[END_REF]. Let U ∈ C ∞ (Ω×[0, 1), R N ) be an extension of u, and let Φ ∈ C 1 c (Ω × [0, 1), R) be an extension of ϕ. Let, for each j ∈ {1, . . . , N + 1}, E j =E j (U) denote the determinant whose columns are the N partial derivatives ∂ 1 U, . . . , ∂ j U,. . . , ∂ N+1 U. By Lemma 1.3, we have

N+1 j=1 (-1) N+ j ∂ j E j = 0.
(2.16)

Combining (2.16) and the divergence theorem, we find that

Ω det (∇u) ϕ = Ω×(0,1) N+1 j=1 ∂ j ((-1) N+ j E j Φ) = Ω×(0,1) N+1 j=1 (-1) N+ j E j ∂ j Φ.
(2.17)

The estimate (2.15) follows from (2.17) via estimates of the quantities E j (U) -E j (V ), where U, respectively V , are well-chosen extensions of u, respectively v.

The Jacobian of sphere-valued maps

The distributional Jacobian plays a major role in the study of maps from

Ω ⊂ R N into S k , 1 ≤ k ≤ N -1.
The analysis of such maps combines analytical, geometrical, and topological tools. We first present, in Sections 3.1-3.3, some typical results in the case where k = N -1, in which geometry plays a moderate role. In Section 3.4, we present the counterparts of these results in the general case 1 ≤ k ≤ N -1. (For more details and further applications, we refer to [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4].)

Vanishing of the Jacobian and approximation by smooth maps

Note that Det (∇u) is well-defined for every u ∈ W 1,N-1 (Ω;

S N-1 ) (since u ∈ L ∞ (Ω; R N )). We claim that Det (∇u) = 0, ∀ u ∈ W 1,N (Ω; S N-1 ). (3.1)
Indeed, on the one hand we know (see Section 2) that

Det (∇u) = det (∇u), ∀ u ∈ W 1,N (Ω; R N ). (3.2)
On the other hand, by differentiating the identity |u| 2 ≡ 1, we find that u • ∂ j u = 0 a.e. on Ω, ∀ j = 1, . . . , N, ∀ u ∈ W 1,1 (Ω; S N-1 ). Thus, for a.e. x ∈ Ω, the vectors ∂ j u(x), j = 1, . . . , N, are contained in a hyperplane, so that det (∇u) = 0 a.e. on Ω. (Alternatively, one could establish (3.1): (i) first, for smooth S N-1 -valued maps; (ii) next, for arbitrary maps u ∈ W 1,N (Ω; S N-1 ), using an approximation argument by smooth S N-1 -valued maps, due to Schoen and Uhlenbeck [START_REF] Schoen | A regularity theory for harmonic maps[END_REF] and relying on the embedding W 1,N ⊂VMO; see also [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4].)

It follows from (3.1) that Det (∇u) = 0, ∀ u ∈ W 1,p (Ω; S N-1 ), ∀ p ≥ N. (3.3)
The restriction p ≥ N in (3.3) is optimal. Indeed, as we are going to see in Section 3.2 below, (3.3) fails when N -1 ≤ p < N. Note however that Det (∇u) = 0, ∀ u ∈ W 1,N-1 (Ω; S N-1 ) ∩ C(Ω; S N-1 ), (3.4) and, more generally,

Det (∇u) = 0, ∀ u ∈ C ∞ (Ω; S N-1 ) W 1,N-1 . (3.5)
Surprisingly, the converse of (3.5) is also true; its proof is non-trivial. More specifically, we have the following

Theorem 3.1. Let N -1 ≤ p < N. Assume that u ∈ W 1,p (Ω; S N-1 ) satisfies Det (∇u) = 0. (3.6) Then u ∈ C ∞ (Ω; S N-1 ) W 1,p . ( 3.7)
Theorem 3.1 is due to Bethuel [START_REF] Bethuel | A characterization of maps in H 1 (B 3 , S 2 ) which can be approximated by smooth maps[END_REF] when p = N -1 and to Bethuel, Coron, Demengel, and Hélein [START_REF] Bethuel | A cohomological criterion for density of smooth maps in Sobolev spaces between two manifolds[END_REF] when N -1 < p < N; see also Ponce and Van Schaftingen [START_REF] Ponce | Closure of smooth maps in W 1,p (B 3 ; S 2 )[END_REF].

The Jacobian as a sum of Dirac masses

We now turn to the example considered in (2.9) and to formula (2.10), which implies in particular that, if u(x) =

xa |x -a| for some a ∈ Ω, then

Det (∇u) = |B N | δ a in D (Ω). (3.8)
We start with an extension of (3.8).

Theorem 3.2. Let u ∈ W 1,N-1 (Ω; S N-1 ) ∩ C(Ω \ {a}). Then Det (∇u) = |B N | deg (u, a) δ a in D (Ω), (3.9)
where deg (u, a) is the (Brouwer) degree of u computed on a small sphere around a. More generally, if u ∈ W 1,N-1 (Ω; S N ) ∩ C(Ω \ {a 1 , . . . , a }), where a 1 , . . . , a ∈ Ω, then

Det (∇u) = |B N | j=1 deg (u, a j ) δ a j in D (Ω).
(3.10) Theorem 3.2 is due to Brezis, Coron, and Lieb [START_REF] Brezis | Harmonic maps with defects[END_REF]. Its proof relies heavily on the Kronecker formula which relates the degree to the Jacobian:

deg f = 1 |S N-1 | S N-1 det (∇ f ), ∀ f ∈ C 1 (S N-1 ; S N-1 ), (3.11)
where det (∇ f ) is the Jacobian of f viewed as a map between manifolds. (See [22, Section 1.1, Section 1.2] for various formulas for deg f and further discussions.) Formula (3.10) implies that the Jacobian of sphere-valued maps conveys information about the location and 'topological strength' of their singularities -a fact that was used and applied in numerous subsequent works.

In the spirit of (3.10), the Jacobian of a general map u ∈ W 1,N-1 (Ω; S N-1 ) is given by the following Theorem 3.3. Let u ∈ W 1,N-1 (Ω; S N-1 ). Then there exist points P j , N j ∈ Ω (not necessarily distinct) such that j

|P j -N j | < ∞ (3.12)
and

Det (∇u) = |B N | j (δ P j -δ N j ) in D (Ω). (3.13)
Conversely, given points P j , N j ∈ Ω satisfying (3.12), there exists some u ∈ W 1,N-1 (Ω; S N-1 ) such that (3.13) holds.

Note that (3.10) and (3.13) are consistent, since, in D (Ω), we may always write j=1 d j δ a j as j (δ P j -δ N j ) for some finite collection of P j 's and N j 's, by repeating the a j 's according to their multiplicity |d j | and adding arbitrary points on ∂Ω. Theorem 3.3 was announced in [START_REF] Brezis | The interplay between analysis and topology in some nonlinear PDE problems[END_REF] and is inspired by an earlier result of Bourgain, Brezis, and Mironescu [9, Theorem 1], [10, Theorem 1] on S 1 -valued maps. It is a special case of Theorem 3.6 in Section 3.4. A sketch of proof of Theorem 3.3 is presented in [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4].

Least energy with prescribed singularities/Jacobian

In this subsection we are concerned with a question originally raised in the context of liquid crystals (see [START_REF] Brezis | Harmonic maps with defects[END_REF] and [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4]): study the least energy S 1 required to generate a configuration having singularities located at given points a j , j = 1, . . . , , in Ω ⊂ R 3 , with given degrees d j ∈ Z, j = 1, . . . , . In a simplified model, the least energy is

S 1 := inf Ω |∇v| 2 ; v ∈ H 1 (Ω; S 2 ) ∩ C(Ω \ {a 1 , . . . , a }) such that deg (v, a j ) = d j , ∀ j . (3.14)
In view of Theorem 3.2, it is reasonable to consider also the quantity

S 2 := inf Ω |∇v| 2 ; v ∈ H 1 (Ω; S 2 ), Det (∇v) = (4π/3) j=1 d j δ a j in D (Ω) . (3.15)
Clearly, S 2 ≤ S 1 , and in fact we can prove that S 2 = S 1 [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Section 4.6]. Going one step further, it is natural to introduce, in full generality, the quantity

S := inf Ω |∇v| N-1 ; v ∈ W 1,N-1 (Ω; S N-1 ), Det (∇v) = |B N | j (δ P j -δ N j )
in D (Ω) , (3.16) where Ω ⊂ R N and the given points P j , N j ∈ Ω satisfy (3.12).

There is a remarkable explicit formula for S. Theorem 3.4. We have S = c N-1 inf j P j -N j ; P j , N j ∈ Ω satisfy (3.12) and j (δ P j -δ N j ) = j (δ P j -δ N j ) in D (Ω) , (3.17) where c N-1 := (N -1) (N-1)/2 |S N-1 |.

In the special case where the collections (P j ), respectively (N j ), consist of a finite number m of points, formula (3.17) becomes

S = c N-1 min σ∈S m m j=1 d(P j , N σ( j) ), (3.18)
where S m denotes the set of permutations of {1, . . . , m} and d is the pseudometric given by d(P, N) := min{|P -N|, dist (P, ∂Ω) + dist (N, ∂Ω}, ∀ P, N ∈ Ω.

Formula (3.18) was established by Brezis, Coron, and Lieb [START_REF] Brezis | Harmonic maps with defects[END_REF] using tools from Optimal Transport. Formula (3.17), which is a generalization of (3.18), is due to Brezis, Mironescu, and Ponce [START_REF] Brezis | W 1,1 -maps with values into S 1 , in Geometric analysis of PDE and several complex variables[END_REF] when N = 2 and Brezis and Mironescu in the general case; a sketch of proof is presented in [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4].

We point out that Theorem 3.4 asserts that the physical quantity S (or rather S 1 ) coincides with a geometrical quantity, i.e., the length (with respect to the pseudometric d) of a 'minimal connection' connecting the points P j to the points N j . Its geometrical nature will become more transparent in the next section, where it will be identified with the least area spanned by a given 'contour' (in the special situation considered in this subsection, the 'contour' corresponds to the points P j , N j ).

Maps with values into lower-dimensional spheres

In this subsection, we consider maps u : Ω → R k+1 (and, in particular, maps with values into S k ⊂ R k+1 ), where Ω ⊂ R N , N ≥ 2, and 1 ≤ k ≤ N -1.

We first introduce the appropriate concept of Jacobian. As in Section 1, it will be convenient to use the language of differential forms, or more precisely the one of -currents, i.e., distributions with values into -forms. (We use below the standard terminology Theorem 3.7. Given any T ∈ F N-k-1 , set

S := inf Ω |∇v| k ; v ∈ W 1,k (Ω; S k ), Jac v = |B k+1 | T . (3.26) Then S = c k inf{|M|; M ⊂ Ω is an (N -k)-rectifiable current such that ∂M = T}, (3.27)
where c

k := k k/2 |S k |.
Here, |M| denotes the mass of M (i.e., its (Nk)-dimensional Hausdorff measure, counted with multiplicities).

We emphasize that Theorem 3.7 asserts that the least 'energy' required to produce singularities on the given 'contour' T coincides, up to a constant, with the least 'surface area' spanned by T (as in the Plateau problem). Theorem 3.7 confirms a conjecture in [START_REF] Brezis | Harmonic maps with defects[END_REF], corresponding roughly speaking to the special case where N = 3, k = 1, and T is a closed curve Γ ⊂ R 3 (as in Theorem 3.5). The conjecture was settled by Almgren, Browder, and Lieb [3] when the singular set is nice. In their remarkable paper, they explained that the upper bound ≤ in (3.27) could be obtained by adapting the dipole construction of [START_REF] Brezis | Harmonic maps with defects[END_REF]; the main new idea in [3] was to point out the importance, in this context, of the coarea formula, that the authors used to derive the lower bound ≥ in (3.27). In full generality, Theorem 3.7 is due to Molnar [START_REF] Molnar | Prescribed singularities with weights[END_REF]; her proof relies heavily on techniques developed in [2] and [3].

Finally, we mention that, in the spirit of Theorem 2.4, one may investigate the existence of the largest fractional Sobolev spaces in which one can define a 'robust' distributional Jacobian, and identify the possible distributional Jacobians arising in this setting. These questions were settled by Bourgain, Brezis, and Mironescu [START_REF] Bourgain | Lifting, degree, and distributional Jacobian revisited[END_REF] when k = N -1, respectively Bousquet and Mironescu [START_REF] Bousquet | Prescribing the Jacobian in critical spaces[END_REF] in the general case.

Improved regularity in

W 1,N (R N ; R N )
We conclude our brief excursion into the enchanted world of Jacobians with several remarkable properties of maps u ∈ W 1,N (Ω; R N ), where N ≥ 2 and Ω ⊂ R N is an open set. Their common feature is the crucial role played by the Jacobian and its algebraic properties.

If u ∈ W 1,N (R N ; R N ), then, clearly, det (∇u) ∈ L 1 (R N ). The following 'microscopic' but rich of consequences improvement of this property is due to Coifman, Lions, Meyer, and Semmes [START_REF] Coifman | Compensated compactness and Hardy spaces[END_REF].

Theorem 4.1. Let u ∈ W 1,N (R N ; R N ). Then det (∇u) ∈ H 1 (R N ).
Here, H 1 (R N ) ⊂ L 1 (R N ) denotes the Hardy space

H 1 (R N ) := { f ∈ L 1 (R N ); ∃ g ∈ L 1 (R N ) such that | f * ϕ ε (x)| ≤ g(x), ∀ x ∈ R N , ∀ ε > 0},
where ϕ is a bump function. (The definition is independent of ϕ.) Slightly smaller than L 1 (R N ), the space H 1 (R N ) is of paramount importance in harmonic analysis (see, e.g., Stein [47,Chapter III]) and enjoys properties that are 'barely' missed by L 1 (R N ), e.g.,

-∆v ∈ H 1 (R N ) =⇒ v ∈ W 2,1 loc (R N ) (4.1)
(see, e.g., [47, Chapter III, Section 3, Theorem 3]).

Another noticeable property of H 1 (R N ) is given by 'Stein's lemma' [START_REF] Stein | Note on the class L log L[END_REF]:

[ f ∈ H 1 (R N ), f ≥ 0 a.e. in Ω] =⇒ [ f ln(2 + f ) ∈ L 1 loc (Ω)]. (4.
2)

The proof of Theorem 4.1 relies on Jacobi's 'fundamental Lemma' 1.1, combined with the following remarkable result, also from [START_REF] Coifman | Compensated compactness and Hardy spaces[END_REF]. Corollary 4.3 is originally due to Wente [START_REF] Wente | An existence theorem for surfaces of constant mean curvature[END_REF]; see also Brezis and Coron [START_REF] Brezis | Multiple solutions of H-systems and Rellich's conjecture[END_REF], where it plays a central role in the solution of Rellich's conjecture. For optimal constants in (4.4), see Topping [START_REF] Topping | The optimal constant in Wente's L ∞ estimate[END_REF] and the references therein. 'Modern' proofs of Corollary 4.3 derive it from Theorem 4.1, (4.1), and the embedding W [START_REF] Brezis | Sur une propriété des espaces de Sobolev[END_REF][START_REF] Brezis | A property of Sobolev spaces[END_REF].

Corollary 4.5, originally due to Müller [START_REF] Müller | Higher integrability of determinants and weak convergence in L 1[END_REF], is a straightforward consequence of Theorem 4.1 and (4.2). For a review of subsequent results in the spirit of Corollary 4.5 (results that do not follow from Theorem 4.1), we refer the reader to the surveys of Iwaniec [START_REF] Iwaniec | Null Lagrangians, the Art of Integration by Parts[END_REF] and Mironescu [START_REF] Mironescu | Le déterminant jacobien[END_REF].

  W s,p (Ω) := {u ∈ L p (Ω); |u| W s,p < ∞}, where |u| W s,p := Ω Ω |u(x) -u(y)| p |x -y| N+sp dxd y 1/p

Theorem 4 . 2 .

 42 Let 1 < p < ∞. If E ∈ L p (R N ; R N ) and B ∈ L p (R N ; R N ) satisfy div E = 0 and curl B = 0 in D (R N ). then E • B ∈ H 1 (R N ).One obtains Theorem 4.1 from Theorem 4.2 by letting p := N/(N -1), and, with the notation of Section 1, E := (C 1, j ) 1≤ j≤N and B := ∇u 1 . Indeed, we havedet (∇u) = N j=1 ∂ j u 1 C 1, j = B • E, and, clearly, B and E satisfy B ∈ L N (R N ; R N ), E ∈ L N/(N-1) (R N ; R N ), curl B = 0 in D (R N )and, by Lemma 1.1, div E = 0 in D (R N ).

Theorem 4 .. 4 ) 4 . 4 .Corollary 4 . 5 .

 444445 1 has three noticeable consequences.Corollary 4.3. Let Ω ⊂ R 2 be a smooth bounded domain. If u ∈ H 1 (Ω; R 2 ), then the problem -∆v = det (∇u) in Ω v = 0 on ∂Ω (4.3) has a (unique) solution v ∈ H 1 0 (Ω) ∩ C(Ω), satisfying ||∇v|| L 2 ≤ C Ω ||∇u 1 || L 2 ||∇u 2 || L 2 and ||v|| L ∞ ≤ C Ω ||∇u 1 || L 2 ||∇u 2 || L 2 . (4Corollary Let Ω ⊂ R N be a smooth bounded domain. If u ∈ W 1,N (Ω; R N ), then det (∇u) ∈ W -1,N/(N-1) (Ω; R) and |〈 f , det (∇u)〉| ≤ C Ω || f || L N ||∇u|| N L N , ∀ f ∈ W 1,N 0 (Ω; R), ∀ u ∈ W 1,N (Ω; R N ).(4.5)If, in addition, det (∇u) ≥ 0 a.e. on Ω, then f det (∇u) ∈ L 1 (Ω) andΩ f det (∇u) = 〈 f , det (∇u)〉, ∀ f ∈ W Let u ∈ W 1,N loc (Ω; R N ).If det (∇u) ≥ 0 a.e. in Ω, then det (∇u) ln(2+det (∇u)) ∈ L 1 loc (Ω).

2 , 1 loc (R 2 )

 212 ⊂ C(R 2 ). Assertion (4.5) in Corollary 4.4 is a consequence of Theorem 4.1, the embedding W 1,N (R N ) ⊂ BMO (R N ) (the John-Nirenberg space of bounded mean oscillation functions), and of Fefferman's theorem asserting that [H 1 (R N )] * = BMO (R N ) [47, Chapter IV]. Assertion (4.6) in Corollary 4.4 relies on a result of Brezis and Browder

about currents. If needed, we refer, e.g., to [START_REF] Brezis | Sobolev Maps to the Circle. From the Perspective of Analysis, Geometry, and Topology[END_REF]Chapter 4] for precise definitions.) Given u ∈ W 1,k (Ω; R k+1 ) ∩ L ∞ (Ω; R k+ 1 ), we set (3.19) where Λ denotes the space of -forms on R N and

(3.20)

For many purposes, it is more convenient to consider, instead of the (k + 1)-current Ju, the (Nk -1)-current Jac u := * Ju (3.21) (where * is the Hodge * operator). This is consistent with Sections 1 and 2, since, when

Our goal is to present far-reaching extensions of the results in Sections 3.1-3.3, involving Jac u when u ∈ W 1,k (Ω; S k ), with arbitrary N > k. We start with an analogue of Theorem 3.2.

Here, Γ is identified, as usual, with an (Nk -1)-current, and m ∈ Z is defined as follows. For any x ∈ Γ, let N = N(x) be the (k + 1)-dimensional normal space to Γ at x, oriented by Γ. Then m is the Brouwer degree of u restricted to the sphere B(x, ε) of N, for sufficiently small ε. (One may prove that the definition does not depend on x or ε.)

In order to generalize Theorem 3.3, it is convenient to introduce a distinguished class of currents. Given an integer with 0 ≤ ≤ N -2, set F := {T; T is an -current such that T = ∂M for some ( + 1) -rectifiable current M on Ω}.

(

The next result provides a complete geometric description of Jac u for a general u ∈ W 1,k (Ω; S k ).

Theorem 3.6. For any u

Conversely, given any T ∈ F N-k-1 , there exists some u ∈ W 1,k (Ω; S k ) such that Jac u = |B k+1 | T.

(3.25) Theorem 3.6, due to Alberti, Baldo, and Orlandi [2] (see also [START_REF] Alberti | Distributional Jacobian and singularities of Sobolev maps[END_REF]), is a far-reaching generalization of Theorem 3.3.

In the general setting considered in this subsection, Theorem 3.4 reads as follows.


Vincent Placet
email: vincent.placet@univ-fcomte.fr

Patrick Perré

A CUSTOM DEVICE ABLE TO MEASURE VISCOELASTIC PROPERTIES OF WOOD UNDER WATER-SATURATED CONDITIONS UP TO 140°C (WAVE T : ENVIRONMENTAL VIBRATION ANALYSER FOR WOOD)

This work presents an innovative experimental device conceived to characterize the time-dependent behavior of hygroscopic materials, as wood, at controlled moisture content and temperature. This device, the WAVE T , permits the determination of the viscoelastic properties of samples, using harmonic tests at frequencies varying from 5.10 -3 Hz to 10 Hz. Conceived to work up to 5 bar, it allows tests in dry or water-saturated conditions to be performed over the temperature range 0°C to 140°C. In spite of these severe working conditions, the careful device design, together with a rigorous data analysis, allows a rigorous determination of the storage and loss modulus and the loss factor. The results collected for several species of wood emphasize the ability of the WAVE T to underline the influence of numerous parameters, namely specie, material direction, anatomical and macromolecular structure, on the rheological properties and notably on the softening temperature. The WAVE T also establishes a efficient tool to follow the modifications of the constitutive amorphous polymers of materials submitted to hydrothermal treatments, in relation to the evolution of their viscoelastic properties.

INTRODUCTION

In comparison with quasi-static methods, harmonic tests represent a more efficient technique to characterize viscoelastic properties and glass transitions of materials. Their major advantage is to cover several decades of time independently of temperature control. In addition, this method allows the viscoelsatic behavior to be distinguished from other effects encountered with wood, such as ageing or growth stress recovery. Consequently, numerous commercial DMA apparatus using this forced nonresonance technique are available. They typically cover wide ranges of frequencies (10 -4 to 200 Hz for the most successful), and are most of the time fitted out of a hot/cold chamber. However, large discrepancies are often observed among data obtained by different dynamic mechanical analyzer, different analysis methods, or even different laboratories using identical instruments (Pournoor and Seferis, [1], Hagen et al., [2]). A number of mechanical factors, like mechanical inertia (resonance frequency), specimen geometry and size, clamping effects may influence the results of dynamic mechanical analysis. Actually, all modes of geometry (flexion, traction, shear, compression…) proposed by commercial apparatus exhibit bias due to imperfect clamping and sample geometry. To make by-pass these problems, the DMA apparatus apply correction factors to extract the viscoelastic properties of samples from the raw data. These corrective factors are fitted for isotropic material, because this type of device is intended for classical polymers. In comparison to these products, materials of vegetable origins, such as wood, present specific features, namely anisotropy and hygroscopicity. The former is problematic concerning the corrections factors and the latter is particularly important because water is recognized to act as very efficient plasticizer in hygroscopic materials. Therefore, in the case of wood, the control of the humidity is absolutely required during dynamic mechanical analysis. Very few DMA apparatus do have humidity control, and even so, it is very difficult to keep the same moisture content of wood when the temperature changes. Notice that the moisture content variations, like drying, may cause collapse and mechano-sorptive effects. One way to avoid these problem is to proceed in a temperature-controlled water bath. The only apparatus which have the possibility to immerse the samples in fluid can only controlled the temperature up to the boiling point of the liquid, that is to say up to near 100°C for water. Even so, an improved knowledge of the rheological properties in water-saturated conditions above 100°C is essential to develop and optimize the processing and manufacturing of wood and raw vegetable materials (drying, wood forming, thermal treatment, veneer, panel pressing, wood gluing and so on…). For all these reasons, the wood rheology team at LERMAB developed an original experimental device, WAVE T , able to perform dynamic mechanical tests of wood under water-saturated conditions and high temperature (up to 140°C with the present version). WAVE T stands for Environmental Vibration Analyzer for Wood.

DESCRIPTION OF THE WAVE T

The WAVE T is able to determine the evolution of the viscoelastic properties of samples with regard to temperature: 5 to 140°C, frequency: 5.10 -3 to 10 Hz and stress level: 0.01 to 4 MPa. It comprises a test bench, a complete chain of vibratory analysis and a conditioning room, which allows the immersion and the temperature control of the sample during experimentation (Fig. 1).

Figure 1. The WAVE T device.

The entire system, composed of the sample holder, the solicitation system and the sensors, is positioned in a pressurized conditioning room. This specific chamber allows the sample to be immersed in water at a temperature level reaching 140°C while maintaining mild conditions around the electronic apparatus (temperature near 20°C and relative humidity around 10%). To carry out the sample solicitation and accurately measure the force and the deflection, non-airtight communicating areas exist between the two compartments.This conditioning system, described in Fig. 1, is patented (Placet and Perré, [START_REF] Placet | Chambre d'essai bi-climatique[END_REF]).

In spite of these severe working conditions, a lot of precautions were carried out in order to ensure the relevance of the system and to allow a rigorous determination of the viscoelastic properties. The harmonic force is applied to the sample using a mini-shaker. In most commercial instruments, the force applied to the sample is deduced from the input signal to the electro-magnetic coils of the driver. In our device, a miniature load cell is used to measure the acual force. As additional advantage, a feedback loop was built with suitable PID parameters, so the force applied to the sample perfectly follows the generator signal. In this way, it is possible to apply the desired sinusoidal force at very low frequency levels: tests have been performed successfully at 1 mHz. The sole limitation is indeed the time required to get experimental data. To ensure rigorous mechanical measurements, a specific articulated clamp was designed to apply a pure vertical force without any momentum to the sample and without any vertical play. Therefore, a pure single cantilever configuration is obtained. Finally, in order to avoid any disturbance due to the clamp area, the deflection measurement is dissociated from the load system.

To keep the advantages of these experimental precautions, great care was taken to collect, treat, and analyze the raw data. The viscoelastic parameters (storage modulus E', loss modulus E'' and loss factor tanδ) are determined taking into consideration of the macroscopic dimensions of the samples at 20°C, the measured force and deflection, and the phase difference. The relevance of the measurements mostly depends on how accurately the phase difference can be obtained. In particular, the method should be able to withdraw the experimental noise level. This is why an inverse method was chosen to extract this phase difference over several periods of the sinusoidal signals. Moreover, to correct or reject measurements done close to the resonance frequency, an analytical model was developed (Fig. 2.). It allows the phase change due to the inertia of all moving parts, the damping effects (water and rubber membrane of the shaker) and the frame stiffness to be dissociated from the sample properties.

All the operations are controlled by a software developed in Visual Basic, which automatically pilots the different apparatus and uses a DLL module (Dynamic Link Library) written in Fortran to identify the parameters. This application also includes the analytical model. The operator has just to key in the desired experimental protocol and to retrieve the file containing the treated data at the end of the experiment.

SOME RESULTS

The accuracy of this apparatus has been tested using standard samples of polymer. For measurements performed on wood samples, a comparative study with a commercial DMA apparatus allowed the advantages of WAVE T to be confirmed (Placet [4]). Several measurement campaigns have been performed on various wood species using the WAVE T apparatus (Placet and Perré, [5,6]). Results revealed a transition in the vicinity of 70 °C to 105°C depending on the wood species, the wood directions and the frequency (Fig. 3a). This transition is related to the glass transition of lignins, one of the three main constitutive polymers of wood (Olsson and Salmén, [START_REF] Olsson | Chap. 9 of Viscoelasticity of Biomaterials[END_REF]). In these tests, it is expected that the temperature level solely acts as a mean of activation of the viscoelastic properties. Nevertheless, other tests, performed at constant temperature over several hours, proved that the temperature also alters the material itself. Contrary to what is generally expected, this effect is significant from 100°C (Fig. 3b).

CONCLUSIONS

During the two last years, by collecting a lot of data in routine on wood samples, the WAVE T achieved a good level of confidence. The collected data and parameters could be used to predict and simulate the behaviour of green wood in drying, forming and peeling processes. The unique ability of WAVE T to assess the thermal degradation during the treatment could also be applied to define the best parameters for the heat treatment of wood in liquid water (temperature/time pathway). The increasing interest to produce ethanol from lignified biomass is certainly a good motivation for this application. In the future, an important improvement will be implemented: a controlled gas supply, based on a mixture of dry gas and saturated gas (Perré et al. 2007, [8]), to accurately control the relative humidity and their variations during the tests. b.

Frequency : 1 Hz Frequency : 1 Hz Tg : 78°C Tg : 94°C

Figure 2 .

 2 Figure 2. Diagram of the comprehensive model and its constitutive equations

Figure 3 .

 3 Figure 3. Evolution of the viscoelastic properties as a function of temperature (a) and time (b) for oak samples, in water saturated conditions and radial direction.


Petru Mironescu

A remark on the distributional Jacobian

Keywords:

We investigate the existence of the distributional Jacobian for maps u ∈ W

In 2D, we present necessary and sufficient existence assumptions. In dimensions 3 and higher, we exhibit sufficient existence conditions weaker than the standard ones.

Introduction

This note complements the expository text [START_REF] Brezis | A brief history of the Jacobian[END_REF] by H. Brezis, J. Mawhin, and the author, presenting a brief -and personal -history of the (distributional) Jacobian. Given a mapping u ∈ W 1,1 loc (Ω; R N ), where N ≥ 2 and Ω ⊂ R N is an open set, and indices 1 ≤ i, j ≤ N, we denote by C i, j the cofactor of ∂ j u i in the Jacobian det (∇u) of u. Note that C i, j is a measurable function defined a.e., but this function need not belong to L 1 loc (Ω; R). Assuming extra regularity assumptions on u, e.g.,

u ∈ W 1,N-1 loc (Ω; R N ) ∩ L ∞ loc (Ω; R N ) (1.1) or u ∈ W 1,N 2 /(N+1) loc (Ω; R N ), (1.2)
we have u i C i, j ∈ L 1 loc (Ω; R), and then we may define

T i := N j=1 ∂ j (u i C i, j ) ∈ D (Ω; R).
(1.3) Moreover, under the assumptions (1.1) or (1.2), T i is independent of i, i.e.,

N j=1 ∂ j (u i C i, j ) = N j=1 ∂ j (u C , j ) in D (Ω; R), ∀ 1 ≤ i, ≤ N, (1.4)
and we may thus define

Det (∇u) := T i , ∀ 1 ≤ i ≤ N, (1.5)
the distributional Jacobian made popular by the seminal work of J. Ball [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF].

The question we address here is the validity of (1.4) for u ∈ W 1,1 loc (Ω; R N ) under 'minimal' extra assumptions, i.e, we look for 'minimal' assumptions allowing us to define the distributional Jacobian as in (1.5). For the sake of simplicity of the statements, we consider only the case i = 1, = 2. Given the local nature of our problem, we may assume that Ω = (-1, 1) N , and discard 'loc'. It will be convenient to use the language of differential forms, and to consider the following more general situation. Let 2 ≤ k ≤ N and

u = (u 1 , . . . , u k ) ∈ W 1,1 (Ω; R k ).
(1.6)

We want to find 'minimal' assumptions implying that

dω 1 + dω 2 = 0 in D (Ω; Λ k ), (1.7)
where Λ k stands for the space of k-forms and

ω 1 := u 1 du 2 ∧ du 3 ∧ . . . ∧ du k , ω 2 := u 2 du 1 ∧ du 3 ∧ . . . ∧ du k (1.8)
are (k -1)-forms with measurable coefficients. (Our original question corresponds to

N = k ≥ 2.)
Note that, formally, (1.7) is clear, since, for smooth u,

ω 1 + ω 2 = d(u 1 u 2 du 3 ∧ . . . ∧ du k ),
and thus

dω 1 + dω 2 = d 2 (u 1 u 2 du 3 ∧ . . . ∧ du k ) = 0.
In order to give a meaning to (1.7), we require

ω 1 , ω 2 ∈ L 1 (Ω; Λ k-1
).

(1.9)

In the easier case k = 2, the minimal conditions (1.6) and (1.9) are sufficient for the validity of (1.7). In fact, more can be said in this case.

Proposition 1. Let N ≥ 1. Let u 1 , u 2 ∈ W 1,1 (Ω; R). Assume that u 1 du 2 + u 2 du 1 ∈ L 1 (Ω; Λ 1 ). Then: 1. u 1 u 2 ∈ W 1,1 (Ω; R). 2. d(u 1 u 2 ) = u 1 du 2 + u 2 du 1 . In particular: (a) We have d(u 1 du 2 + u 2 du 1 ) = 0 in D (Ω; Λ 2 ). (b) Under the stronger assumptions u 1 du 2 ∈ L 1 (Ω; Λ 1 ) and u 2 du 1 ∈ L 1 (Ω; Λ 1 ), we have d(u 1 du 2 ) + d(u 2 du 1 ) = 0 in D (Ω; Λ 2 ). Consequently, (1.7) with k = 2
holds under the assumptions (1.6) and (1.9).

||u

1 u 2 || L 1 ≤ C(||u 1 || W 1,1 ||u 2 || W 1,1 + ||u 1 du 2 + u 2 du 1 || L 1 ) (with C depending only on N). Corollary 1. Let Ω ⊂ R 2 be an open set. The distributional Jacobian Det (∇u) is well- defined (via (1.5)) for maps u = (u 1 , u 2 ) : Ω → R 2 satisfying u 1 , u 2 ∈ W 1,1 loc (Ω; R), u 1 du 2 , u 2 du 1 ∈ L 1 loc (Ω; Λ 1 ).
When k ≥ 3, the necessary conditions (1.6) and (1.9) are no more sufficient for the validity of (1.7).

Proposition 2. Let N ≥ k ≥ 3. There exists some u = (u 1 , . . . , u k ) ∈ W 1,1 (Ω; R k ) such that: a) u j ∈ C ∞ (Ω; R), j = 2, 3. b) u j ∈ W 1,1 (Ω; R) ∩ L ∞ and du j ∈ L 2,∞ (Ω; Λ 1 ), j = 2, 3. (So that u 2 and u 3 are 'almost' in W 1,2 .) c) ω 1 , ω 2 ∈ L 2,∞ (Ω; Λ k-1 ). d) dω 1 + dω 2 = 0 in D (Ω; Λ k ).
Finally, we present an existence result for the distributional Jacobian when k ≥ 3. Motivated by the example in Proposition 2 (see also Remark 1 in the next section) and the validity of (1.7) under the assumption (1.1), we consider the following assumption, weaker than (1.1):

du j ∈ L k-1 ([ε < |u j | < M]), ∀ 0 < ε < M < ∞, ∀ 1 ≤ j ≤ k.
(1.10) Proposition 3. Assume (1.6), (1.9), and (1.10). Then (1.7) holds.

Corollary 2. Let N ≥ 3 and let Ω ⊂ R N be an open set. Let u ∈ W 1,1 loc (Ω; R N ) satisfy |u| |du| N-1 ∈ L 1 (Ω; R) (1.11)
(where | | stands for the Euclidean length). Then the distributional Jacobian Det (∇u) is well-defined via (1.5).

Acknowldegment. I warmly thank H. Brezis for useful discussions during the writing of [START_REF] Brezis | A brief history of the Jacobian[END_REF], and in particular for suggesting (1.11) as a sufficient condition for the existence of the distributional Jacobian.

Proofs

Proof of Proposition 1. In what follows, C denotes a generic constant depending only on N.

When N = 1, the assumption u 1 u 2 + u 2 u 1 ∈ L 1 is always satisfied, part 2 is simply the chain rule (u 1 u 2 ) = u 1 u 2 + u 2 u 1 , and the estimate in part 3 follows from the embedding

W 1,1 → L ∞ .
Let N ≥ 2. Using the case N = 1 and Fubini, we find that parts 1 and 2 hold provided we know that u 1 u 2 ∈ L 1 .

In particular, when N = 2, parts 1, 2, and 3 follow from the above considerations and the embedding W 1,1 → L 2 , which implies that

||u 1 u 2 || L 1 ≤ C||u 1 || W 1,1 ||u 2 || W 1,1 .
The case N ≥ 3 follows by induction on N. With no loss of generality, we may assume that u 1 and u 2 are absolutely continuous on a.e. line segment of the form

L (x 1 ,...,x N-1 ) := {(x 1 , . . . , x N-1 , t); -1 < t < 1}.
Pick some t 0 ∈ (-1, 1) such that the partial functions (-1, 1) N-1 (x 1 , . . . , x N-1 ) → v i (x 1 , . . . , x N-1 ) := u i (x 1 , . . . , x N-1 , t 0 ), i = 1, 2, satisfy v 1 , v 2 ∈ W 1,1 ((-1, 1) N-1 ; R) and the estimates

||v i || W 1,1 ≤ C||u i || W 1,1 , i = 1, 2,
(2.1)

||v 1 dv 2 + v 2 dv 1 || L 1 ≤ C||u 1 du 2 + u 2 du 1 || L 1 . (2.2)
By (2.1), (2.2), and the induction assumption, we then have v 1 v 2 ∈ L 1 ((-1, 1) N-1 ; R), and the estimate

||v 1 v 2 || L 1 ≤ C(||u 1 || W 1,1 ||u 2 || W 1,1 + ||u 1 du 2 + u 2 du 1 || L 1 ).
(2.3)

On the other hand, for a.e. (x 1 , . . . , x N-1 ) ∈ (-1, 1) N-1 , the Leibniz rule applies, on L (x 1 ,...,x N-1 ) , to the absolutely continuous partial maps

(-1, 1) t → w i (t) := u i (x 1 , . . . , x N-1 , t), i = 1, 2,
which implies, for such (x 1 , . . . , x N-1 ), that

||w 1 w 2 || L 1 ((-1,1)) ≤ C(|w 1 (t 0 ) w 2 (t 0 )| + w 1 w 2 + w 2 w 1 L 1 ((-1,1)) ).
(2.4) Combining (2.3) and (2.4), we find that item 3 holds, and in particular, that u 1 u 2 ∈ L 1 (Ω; R). As explained above, this also implies the validity of items 1 and 2.

Proof of Proposition 2. Set, for x ∈ Ω, ρ = ρ(x) := (x 2 1 + x 2 2 ) 1/2 , and let

u 1 (x) := 1, u 2 (x) := x 1 ρ e x 3 , u 3 (x) := x 2 ρ e x 3 , u j (x) := x j , 4 ≤ j ≤ k.
Properties a) and b) are straightforward.

We now prove c). Clearly, ω 2 = 0. On the other hand, we have

du 2 = x 2 2 ρ 3 e x 3 dx 1 - x 1 x 2 ρ 3 e x 3 dx 2 +
x 1 ρ e x 3 dx 3 in D (Ω; Λ 1 ),

du 3 = - x 1 x 2 ρ 3 e x 3 dx 1 + x 2 1 ρ 3 e x 3 dx 2 +
x 2 ρ e x 3 dx 3 in D (Ω; Λ 1 ), and therefore, with ω 0 := dx 4 ∧ . . . ∧ dx k (and the convention ω 0 := 1 if k = 3)

ω 1 = - x 1 ρ 2 e 2x 3 dx 2 ∧ dx 3 + x 2 ρ 2 dx 3 ∧ dx 1 ∧ ω 0 in D (Ω; Λ k-1
).

(2.5) Thus c) holds.

Finally, using (2.5) and the fact that

∂ 1 x 1 ρ 2 + ∂ 2 x 2 ρ 2 = 2π δ (0,0) in D (R 2 ; R),
we find that

dω 1 + dω 2 = dω 1 = -2π δ (0,0) ⊗ e 2x 3 dx 1 ∧ • • • ∧ dx k in D (Ω; Λ k ),
and thus d) holds.

Remark 1. A similar example shows that, in order to insure the validity of (1.4), the condition (1.1) cannot be weakened to u j ∈ L ∞ loc (Ω; R) and du j ∈ L N-1,∞ loc

(Ω; Λ 1 ), 1 ≤ j ≤ N. Indeed, if we set r = r(x) := (x 2 1 + • • • + x 2 k-1 ) 1/2 and let u 1 (x) := 1, u j (x) := x j-1 r e x k , 2 ≤ j ≤ k,
then (by repeating the proof of Proposition 2) we have

u j ∈ L ∞ (Ω; R), du j ∈ L k-1,∞ (Ω; Λ 1 ), 1 ≤ j ≤ k, ω 1 , ω 2 ∈ L k-1,∞ (Ω; Λ k-1
), and

dω 1 + dω 2 = 0 in D (Ω; Λ k ). Proof of Proposition 3. Consider a sequence (Φ ) ≥2 ⊂ C ∞ (R; R) such that Φ (t) = 0 if |t| < 1/(2 ), Φ (t) = ( + 1) sgn t if |t| > + 1, Φ (t) = t if 1/ ≤ |t| ≤ , and |Φ | ≤ 2, ∀ . Set u j = u j, := Φ (u j ).
The strategy of the proof consists of proving (1.7) for u j instead of u j , and then letting → ∞.

Since Indeed, (2.7) follows by approximation with smooth maps, using the fact that, by the assumption (1.10) and the chain and Leibniz rules, we have u 1 u 2 , u 3 , . . . , u k ∈ W 1,k-1 .

u 1 , u 2 ∈ W 1,1 ∩ L ∞ , the Leibniz rule implies that d( u 1 u 2 ) = u 1 d u 2 + u 2 d u 1 ,
Combining (2.6) and (2.7), we find that (2.8)

We obtain (1.7) by letting → ∞ in (2.8). (Recall that u j = u j, .) Passing to the limits is justified via the assumption (1.9) and the uniform inequalities

  and thus( u 1 d u 2 + u 2 d u 1 ) ∧ d u 3 ∧ . . . ∧ d u k = d( u 1 u 2 ) ∧ d u 3 ∧ . . . ∧ d u k in D (Ω; Λ k-1 ). (2.6)We claim that, under the only assumption (1.10), we haved( u 1 u 2 ) ∧ d u 3 ∧ . . . ∧ d u k = d(( u 1 u 2 ) d u 3 ∧ • • • ∧ d u k )) in D (Ω; Λ k ) (2.7)(and this also holds for N = k -1).

d( u 1 d u 2 ∧

 2 d u 3 ∧ . . . ∧ d u k ) + d( u 2 d u 1 ∧ d u 3 ∧ . . . ∧ d u k ) = 0 in D (Ω; Λ k ).