V Bonnaillie-No

S Fournais
email: fournais@imf.au.dk

SUPERCONDUCTIVITY IN DOMAINS WITH CORNERS

We study the two-dimensional Ginzburg-Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large κ regime. Furthermore, we discuss nucleation of superconductivity at the boundary.

It is a well-known phenomenon that superconductors of Type II lose their superconducting properties when submitted to sufficiently strong external fields. The value of the external field where this transition takes place is usually called H C3 , and is calculated as a function of a material-dependent parameter κ. The calculation of this critical field, H C3 , for large values of κ has been the focus of much activity [BeSt], [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF][START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF][START_REF] Lu | Gauge invariant eigenvalue problems on R 2 and R 2 +[END_REF], [PiFeSt], [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] and [HePa]. In the recent works [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF][START_REF] Fournais | Strong diamagnetism for general domains and applications[END_REF] the definition of H C3 in the case of samples of smooth cross section was clarified and it was realized that the critical field is determined completely by a linear eigenvalue problem. The linear spectral problem has been studied in depth in the case of corners in [START_REF] Bonnaillie | Analyse mathématique de la supraconductivité dans un domaine à coins : méthodes semi-classiques et numériques[END_REF][START_REF] Bonnaillie | On the fundamental state for a Schrödinger operator with magnetic fields in domains with corners[END_REF][START_REF] Bonnaillie-Noël | Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners[END_REF]. The objective of the present paper is to use the spectral information from [BonDa] to carry through an analysis similar to the one in [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF] in the case of corners. Thereby we will in particular obtain: 1) A complete asymptotics of H C3 for large values of κ in terms of linear spectral data, 2) Precise estimates on the location of nucleation of superconductivity for magnetic field strengths just below the critical field. The case of corners of angle π/2 has been studied in [START_REF] Jadallah | The onset of superconductivity in a domain with a corner[END_REF][START_REF] Pan | Upper critical field for superconductors with edges and corners[END_REF]. Our results are more precise-even for those angles-and we study more general domains.

We will work in the Ginzburg-Landau model. Let Ω ⊂ R 2 be a bounded simply connected domain with Lipschitz boundary. The Ginzburg-Landau functional is given by

E[ψ, A] = E κ,H [ψ, A] = Ω |p κHA ψ| 2 -κ 2 |ψ| 2 + κ 2 2 |ψ| 4 dx + κ 2 H 2 R 2 |curl A -1| 2 dx , (1.1)
with ψ ∈ W 1,2 (Ω; C), A in the space Ḣ1 F,div that we will define below, and where p A = (-i∇ -A). Notice that the second integral in (1.1) is over the entire space, R 2 , whereas the first integral is only over the domain Ω.

Formally the functional is gauge invariant. In order to fix the gauge, we will impose that vector fields A have vanishing divergence. Therefore, a good choice for the variational space for A is

Ḣ1 F,div = F + Ḣ1 div , (1.2)
where

Ḣ1 div = {A ∈ Ḣ1 (R 2 , R 2 ) div A = 0}
. Furthermore F is the vector potential giving constant magnetic field F(x 1 , x 2 ) = 1 2 (-x 2 , x 1 ), (1.3) and we use the notation Ḣ1 (R 2 ) for the homogeneous Sobolev spaces, i.e. the closure of C ∞ 0 (R 2 ) under the norm

f → f Ḣ1 = ∇f L 2 .
Any square integrable magnetic field B(x) can be represented by a vector field A ∈ Ḣ1 div . Minimizers, (ψ, A) ∈ W 1,2 (Ω) × Ḣ1 F,div , of the functional E have to satisfy the Euler-Lagrange equations:

p 2 κHA ψ = κ 2 (1 -|ψ| 2 )ψ in Ω , (1.4a) curl 2 A = -i 2κH (ψ∇ψ -ψ∇ψ) -|ψ| 2 A 1 Ω (x) in R 2 , (1.4b) (p κHA ψ) • ν = 0 on ∂Ω . (1.4c)
It is standard to prove that for all κ, H > 0, the functional E κ,H has a minimizer. An important result by Giorgi and Phillips, [GiPh], states that for κ fixed and H sufficiently large (depending on κ), the unique solution of (1.4) (up to change of gauge) is the pair (ψ, A) = (0, F). Since ψ is a measure of the superconducting properties of the state of the material and A is the corresponding configuration of the magnetic vector potential, the result of Giorgi and Phillips reflects the experimental fact that superconductivity is destroyed in a strong external magnetic field.

We define the lower critical field H C3 as the value of H where this transition takes place:

H C3 (κ) = inf{H > 0 : (0, F) is a minimizer of E κ,H } .
(1.5) However, it is far from obvious from the functional that the transition takes place at a unique value of H-there could be a series of transitions back and forth before the material settles definitely for the normal state, (0, F). Therefore, we introduce a corresponding upper critical field

H C3 (κ) = inf{H > 0 : (0, F) is the unique minimizer of E κ,H ′ for all H ′ > H} . (1.6)
Part of our first result, Theorem 1.4 below, is that the above definitions coincide for large κ.

Let us introduce some spectral problems. For B ≥ 0 and a (sufficiently regular) domain Ω ⊂ R 2 , we can define a quadratic form

Q[u] = Q Ω,B [u] = Ω |(-i∇ -BF)u| 2 dx, (1.7) with form domain {u ∈ L 2 (Ω) (-i∇ -BF)u ∈ L 2 (Ω)}.
The self-adjoint operator associated to this closed quadratic form will be denoted by H(B) = H Ω (B). Notice that since the form domain is maximal, the operator H Ω (B) will correspond to Neumann boundary conditions. We will denote the n'th eigenvalue of H(B) (counted with multiplicity) by λ n (B) = λ n,Ω (B), in particular,

λ 1 (B) = λ 1,Ω (B) := inf Spec H Ω (B).
The case where Ω is an angular sector in the plane will provide important special models for us. Define, for 0 < α ≤ 2π,

Γ α := {z = r(cos θ, sin θ) ∈ R 2 r ∈ (0, ∞), |θ| < α/2}.
Since this domain is scale invariant one easily proves that Spec H Γα (B) = B Spec H Γα (1). Therefore, we set B = 1 and define

µ 1 (α) = λ 1,Γα (B = 1).
(1.8)

The special case of α = π, i.e. the half plane, has been studied intensively. In compliance with standard notation, we therefore also write

Θ 0 := µ 1 (α = π).
It is known that the numerical value of Θ 0 is Θ 0 = 0.59..... Remark 1.1.

It is believed-and numerical evidence exists (cf. [AlBo, BDMV] and Figure 1) to support this claim-that α → µ 1 (α) is a strictly increasing function on [0, π] and constant equal to Θ 0 on [π, 2π]. If this belief is proved, then the statement of our Assumption 1.3 below can be made somewhat more elegantly.

We consider Ω a domain whose boundary is a curvilinear polygon in the sense given by Grisvard, see Definition 1.2. Definition 1.2 (cf. [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]).

Let Ω be a bounded open subset of R 2 . We say that the boundary Γ is a (smooth) curvilinear polygon, if for every x ∈ Γ there exists a neighborhood V of x in R 2 and a mapping ψ from V in R 2 such that

(1) ψ is injective, (2) ψ together with ψ -1 (defined on ψ(V )) belongs to the class C ∞ , (3) Ω ∩ V is either {y ∈ Ω | ψ 2 (y) < 0}, {y ∈ Ω | ψ 1 (y) < 0 and ψ 2 (y) < 0}, or {y ∈ Ω | ψ 1 (y) < 0 or ψ 2 (y) < 0}
, where ψ j denotes the components of ψ.

From now on, we consider a bounded open subset Ω ⊂ R 2 , whose boundary is a curvilinear polygon of class C ∞ . The boundary of such a domain will be a piecewise smooth curve Γ. We denote the (minimal family of) smooth curves which make up the boundary by Γ j for j = 1, . . . , N . The curve Γ j+1 follows Γ j according to a positive orientation, on each connected component of Γ. We denote by s j the vertex which is the end point of Γ j . We define a vector field ν j on a neighborhood of Ω, which is the unit normal a.e. on Γ j .

We will work under the following assumption on the domain.

Assumption 1.3.

The domain Ω has curvilinear polygon boundary and denote the set of vertices by Σ. We suppose that N := |Σ| = 0. We denote by α s the angle at the vortex s (measured towards the interior). We suppose that µ 1 (α s ) < Θ 0 for all s ∈ Σ, and define Λ 1 := min s∈Σ µ 1 (α s ). We also assume that α s ∈ (0, π) for all s ∈ Σ.

Under this assumption we resolve the ambiguity of definition of H C3 (κ) and derive a complete asymptotics in terms of spectral data.

Theorem 1.4. Suppose that Ω is a bounded, simply-connected domain satisfying Assumption 1.3.

Then there exists κ 0 > 0 such that if κ ≥ κ 0 then the equation λ 1,Ω (κH) = κ 2 , has a unique solution H = H lin C3 (κ). Furthermore, if κ 0 is chosen sufficiently large, then for κ ≥ κ 0 , the critical fields defined in (1.5), (1.6) coincide and satisfy

H C3 (κ) = H C3 (κ) = H lin C3 (κ).
(1.9)

Finally, the critical field has a complete asymptotic expansion in powers of κ -1 : There exists {η j } ∞ j=1 ⊂ R such that

H C3 (κ) = κ Λ 1 1 + ∞ j=1 η j κ -j , for κ → ∞, (1.10)
in the sense of asymptotic series.

Remark 1.5.

The result analogous to Theorem 1.4 for smooth domains (i.e. for Σ = ∅) has been established in [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF][START_REF] Fournais | Strong diamagnetism for general domains and applications[END_REF]. Notice however that the form of the asymptotics (1.10) depends on the existence of a vortex and is more complicated in the case of smooth domains.

Once Theorem 1.4 is established it makes sense, for large values of κ, to talk of the critical field that we will denote by

H C3 (κ) (= H C3 (κ) = H C3 (κ)).
In the case of regular domains (without corners) one has the asymptotics (see [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF], [PiFeSt], [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] and [HePa]),

H C3 (κ) = κ Θ 0 + O(1),
where the leading correction depends on the maximal curvature of the boundary. We observe that the corners-which can be seen as points where the curvature is infinite-change the leading order term of H C3 (κ). Thus there is a large parameter regime of magnetic field strengths, κ/Θ 0 ≪ H ≤ H C3 (κ), where superconductivity in the sample must be dominated by the corners. Our next two results make this statement precise. First we prove Agmon type estimates, for the minimizers of the non-linear Ginzburg-Landau functional, which describe how superconductivity can nucleate successively in the corners, ordered according to their spectral parameter µ 1 (α s ).

Theorem 1.6. Suppose that Ω satisfies Assumption 1.3, let µ > 0 satisfy min s∈Σ µ 1 (α s ) < µ < Θ 0 and define

Σ ′ := {s ∈ Σ µ 1 (α s ) ≤ µ}. There exist constants κ 0 , M, C, ǫ > 0 such that if κ ≥ κ 0 , H κ ≥ µ -1 , and (ψ, A) is a minimizer of E κ,H , then Ω e ǫ √ κHdist(x,Σ ′ ) |ψ(x)| 2 + 1 κH |p κHA ψ(x)| 2 dx ≤ C {x: √ κHdist(x,Σ ′ )≤M } |ψ(x)| 2 dx.
Finally we discuss leading order energy asymptotics in the parameter regime dominated by the corners, i.e. κ/Θ 0 ≪ H ≤ H C3 (κ). The result below, Theorem 1.7, can be seen as a partial converse to Theorem 1.6 in that all corners which are spectrally permitted will contribute to the leading order of the ground state energy.

One can imagine an interaction between corners with the same spectral parameter, i.e. with the same angle α. This would be a tunnelling type effect and has much lower order. We refrain from a detailed study of such an interaction, since that would be far out of the scope of the present paper.

The ground state energy will be given to leading order by decoupled model problems in angular sectors. It may be slightly surprising to notice that these model problems remain non-linear.

Let α ∈ (0, π) be such that µ 1 (α) < Θ 0 . (Remember that it follows from [Bon2] that µ 1 (α) < Θ 0 for α ∈ (0, π 2 ] and that numerical evidence suggests this to be the case in the entire interval α ∈ (0, π).)

Define, for µ 1 , µ 2 > 0, the following functional J α µ1,µ2 ,

J α µ1,µ2 [ψ] = Γα |(-i∇ -F)ψ| 2 -µ 1 |ψ| 2 + µ 2 2 |ψ| 4 dx, (1.11) with domain {ψ ∈ L 2 (Γ α ) | (-i∇ -F)ψ ∈ L 2 (Γ α )}. Define also the corresponding ground state energy E α µ1,µ2 := inf J α µ1,µ2 [ψ].
The main result on the ground state energy of the Ginzburg-Landau functional in the parameter regime dominated by the corners is the following. H(κ) .

Theorem 1.7. Suppose κ H(κ) → µ ∈ R + as κ → ∞, where µ < Θ 0 . Let (ψ, A) = (ψ, A) κ,H(κ) be a minimizer of E κ,
Then

E κ,H(κ) [ψ, A] → s∈Σ E αs µ,µ , (1.12)
as κ → ∞. Remark 1.8. Proposition 6.1 below states that E αs µ,µ = 0 unless µ 1 (α s ) < µ, so only corners satisfying this spectral condition contribute to the ground state energy in agreement with the localization estimate from Theorem 1.6.

Spectral analysis of the linear problem

2.1. Monotonicity of λ 1 (B).
In this subsection we will prove that B → λ 1 (B) is increasing for large B. Thereby we will have proved the first statement of Theorem 1.4 (see Propositions 2.3 and 2.4 below). Furthermore, Lemma 2.5 establishes the form of the asymptotics of H lin C3 (κ).

In [BonDa] the asymptotics of λ 1 (B) was effectively calculated to any order. Let us recall their results.

Definition 2.1.

Let Ω be a bounded curvilinear polygon. We denote by

• Λ n the n-th eigenvalue of the model operator ⊕ s∈Σ Q αs where Q αs is the magnetic Neumann Laplacian (-i∇ -F) 2 on the infinite angular sector of opening α s . In particular,

Λ 1 = min s∈Σ µ 1 (α s ), • K Ω the largest integer K such that Λ K < Θ 0 , • µ (n) (h) the n-th smallest eigenvalue of the magnetic Neumann Laplacian (-ih∇ -F) 2 on Ω.
Theorem 2.2 ([BonDa] Theorem 7.1).

Let n ≤ K Ω . There exists h 0 > 0 and (m j ) j≥1 such that for any N > 0 and h ≤ h 0 ,

µ (n) (h) = hΛ n + h N j=1 m j h j/2 + O(h N +1
2 ).

Furthermore, if Ω is a bounded convex polygon (i.e. has straight edges), then for any n ≤ K Ω , there exists r n > 0 and for any ε > 0, C ε > 0 such that

µ (n) (h) -hΛ n ≤ C ε exp - 1 √ h (r n Θ 0 -Λ n -ε) .
Recall the notation H(B), λ n (B) introduced after (1.7). By a simple scaling, we get

λ n (B) = B 2 µ (n) (B -1 ), ∀n.
(2.1)

Let us make more precise the behavior of λ 1 (B) as B is large. For this, we define the left and right derivatives of λ 1 (B):

λ ′ 1,± (B) := lim ε→0 ± λ 1 (B + ε) -λ 1 (B) ε . (2.2) Proposition 2.3.
The limits of λ ′ 1,+ (B) and λ ′ 1,-(B) as B → +∞ exist, are equal and we have

lim B→+∞ λ ′ 1,+ (B) = lim B→+∞ λ ′ 1,-(B) = Λ 1 . Therefore, B → λ 1 (B) is strictly increasing for large B.
Proof. This proof is similar to that of [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF].

Let B ≥ 0 and let n be the degeneracy of λ 1 (B). There exist ε > 0, 2n analytic functions φ j and

E j , j = 1, . . . , n defined from (B -ε, B + ε) into H 2 (Ω) \ {0} and R respectively, such that H(β)φ j (β) = E j (β)φ j (β), E j (B) = λ 1 (B),
and such that {φ j (B)} are linearly independent. If ε is small enough, there exist j + and j -in {1, . . . , n} such that

for β ∈ (B, B + ε), E j,+ (β) = min j∈{1,...,n} E j (β), for β ∈ (B -ε, B), E j,-(β) = min j∈{1,...,n} E j (β).
By first order perturbation theory, the derivatives λ ′ 1,± (B) can be rewritten

λ ′ 1,± (B) = -2ℜ φ j± (B), F • (-i∇ -BF)φ j± (B) .
We deduce, for ε > 0,

λ ′ 1,+ (B) = 1 ε φ j+ (B), (H(B + ε) -H(B) -ε 2 F 2 )φ j+ (B) ≥ λ 1 (B + ε) -λ 1 (B) ε -ε F 2 L ∞ (Ω) .
Using Theorem 2.2, we deduce that

λ ′ 1,+ (B) ≥ Λ 1 + m 1 √ B + ε - √ B ε + ε -1 O(B -1/2 ) -ε F 2 L ∞ (Ω) .
Thus, lim inf

B→∞ λ ′ 1,+ (B) ≥ Λ 1 -ε F 2 L ∞ (Ω) .
Since ε is arbitrary, we have lim inf

B→∞ λ ′ 1,+ (B) ≥ Λ 1 .
Taking ε < 0, we obtain by a similar argument,

lim inf B→∞ λ ′ 1,-(B) ≤ Λ 1 .
The two last inequalities and the relation

λ ′ 1,+ (B) ≤ λ ′ 1,-(B) achieve the proof.
We are now able to prove the following proposition.

Proposition 2.4.

The equation in H λ 1 (κH) = κ 2 has a unique solution H(κ) for κ large enough. Proof.

According to Proposition 2.3, there exists B 0 > 0 such that λ 1 is a strictly increasing continuous function from [B 0 , +∞) onto [λ 1 (B 0 ), +∞). By choosing B 0 sufficiently large, we may assume that λ 1 (B) < λ 1 (B 0 ) for all B < B 0 . Let κ 0 = λ 1 (B 0 ), then, for any B > B 0 , the equation

λ 1 (κH) = κ 2 has a unique solution H = λ -1 1 (κ 2 )/κ with λ -1 1 the inverse function of λ 1 defined on [λ 1 (B 0 ), +∞). Lemma 2.5. Let H = H lin C3 (κ) be the solution to the equation λ 1 (κH) = κ 2
given by Proposition 2.4. Then there exists a real valued sequence (η j ) j≥1 such that

H lin C3 (κ) = κ Λ 1 1 + ∞ j=1 η j κ -j , (2.3)
(in the sense of asymptotic series) with Λ 1 = min s∈Σ µ 1 (α s ) introduced in Definition 2.1.

Proof.

By Theorem 2.2 and (2.1) there exists a sequence (m k ) k≥1 such that, for any N ∈ N,

λ 1 (B) = Λ 1 B + B N k=1 m k B -k/2 + O(B -N +1 2
) as B → +∞.

(2.4)

We compute with the Ansatz for H(κ) given by (2.3) :

λ 1 (κH) ∼ Λ 1 κH + κH k≥1 m k (κH) -k/2 ∼ κ 2 1 + ∞ j=1 η j κ -j + k≥1 m k κ 2-k Λ 1-k/2 1 1 + ∞ j=1 η j κ -j 1-k/2 = κ 2 + η 1 + m 1 √ Λ 1 κ + η 2 + m 1 √ Λ 1 η 1 2 + m 2 + . . . = κ 2 + κ 2 j≥1 (η j + mj )κ -j ,
where the coefficients mj only depend on the η k for k < j. Thus, the form (2.3) admits a solution in the sense of asymptotic series. It is an easy exercise to prove that H lin C3 (κ) is equivalent to this series. 2.2. Agmon estimates near corners for the linear problem.

If φ ∈ C ∞ 0 (Ω) (i.e. with support away from ∂Ω) it is a simple calculation to prove that Ω |(-i∇ -A)φ| 2 dx ≥ Ω curl A|φ| 2 dx.
(2.5)

In particular, for A = BF,

Q Ω,B [φ] ≥ B φ 2 . (2.6)
Using the technique of Agmon estimates ( [Ag, Hel]) one can combine the upper and lower bounds (2.4) and (2.6) to obtain exponential localization near the boundary for ground state eigenfunctions of H(B). For completeness we give the following theorem (without proof-we will give the proof of similar non-linear estimates below), though we will not need the result here. Theorem 2.6. Let ψ B be the ground state eigenfunction of H(B). Then there exist constants ǫ, C, B 0 > 0 such that

e ǫ √ Bdist(x,∂Ω) |ψ B (x)| 2 + B -1 |p BF ψ B (x)| 2 dx ≤ C ψ B 2 2 ,
for all B ≥ B 0 .

In order to prove exponential localization near the corners for minimizers of E κ,H we will need the operator inequality (2.7) below (compare to (2.6)).

Theorem 2.7. Let δ > 0. Then there exist constants M 0 , B 0 > 0 such that if B ≥ B 0 then H(B) satisfies the operator inequality

H(B) ≥ U B , (2.7)
where U B is the potential given by

U B (x) :=      (µ 1 (α s ) -δ)B, dist(x, s) ≤ M 0 / √ B, (Θ 0 -δ)B, dist(x, Σ) > M 0 / √ B, dist(x, ∂Ω) ≤ M 0 / √ B, (1 -δ)B, dist(x, ∂Ω) > M 0 / √ B. Proof of Theorem 2.7. Let χ 1 ∈ C ∞ (R) be non-increasing and satisfy χ 1 (t) = 1 for t ≤ 1, χ 1 (t) = 0 for t ≥ 2.
Define, for L, M, B > 0,

χ cor M (x) := χ 1 ( √ Bdist(x, Σ)/M ), χ bd M (x) := (1 -χ 2 1 ) √ Bdist(x, Σ)/M × χ 1 √ BLdist(x, ∂Ω)/(2M ) χ int M (x) := (1 -χ 2 1 ) √ BLdist(x, ∂Ω)/(2M ) .
The parameter L will be fixed. It is chosen sufficiently large that supp χ bd M consists of N (the number of smooth boundary curves) disjoint components (lying along each smooth boundary piece) when √ B/M is large. Using the IMS-formula we can write for any φ ∈ H 1 (Ω),

Q Ω,B [φ] ≥ Q Ω,B [χ cor M φ] + Q Ω,B [χ bd M φ] + Q Ω,B [χ int M φ] -C B M 2 φ 2 , (2.8)
for some constant C > 0 independent of M , B and φ.

We will estimate each term of (2.8) by using successively results for the first eigenvalue of the Schrödinger operator in a domain with one corner, in a smooth domain and in the entire plane.

Since χ int M φ has compact support in Ω, we get (see (2.6))

Q Ω,B [χ int M φ] = Q R 2 ,B [χ int M φ] ≥ B χ int M φ 2 .
(2.9)

For the corner contribution and boundary contribution, we will use the estimates in angular sectors and regular domains obtained in [START_REF] Bonnaillie | On the fundamental state for a Schrödinger operator with magnetic fields in domains with corners[END_REF][START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]. For any corner s ∈ Σ, we define a domain Ω s such that Ω ∩ B(s, ε) = Ω s ∩ B(s, ε) for ε small enough (ε < dist(s, Σ \ {s})) and its boundary is C ∞ except in s. Let s - and s + be the neighbor vertices of s (if they exist). We define two regular domains Ω - s and Ω + s such that there exists ε > 0 with Ω ∩ B(x, ε) = Ω ± s ∩ B(x, ε) for any x ∈ {y ∈ Γ s,s± , ℓ(s, y) ≤ 2/3ℓ(s, s ± )} where Γ s,s± denotes the piece of the boundary of Ω which joins the edges s and s ± and ℓ(s, s ± ) is the length of Γ s,s± . Figures 2 and3 give examples of domains Ω s and Ω ± s . As soon as B/M 2 is large enough, the support of χ cor M is the union of N disjoint domains localized near each corner s, s ∈ Σ. Consequently, for B ≥ B 0 , we can rewrite χ cor M as

χ cor M = s∈Σ χ cor,s M with s ∈ suppχ cor,s M , suppχ cor,s M ∩ suppχ cor,s ′ M = ∅, ∀s = s ′ .
Furthermore, we choose B 0 large enough such that for any B ≥ B 0 ,

suppχ cor,s M ∩ Ω ⊂ Ω s , ∀s ∈ Σ.
Using the eigenvalue asymptotics from [START_REF] Bonnaillie | On the fundamental state for a Schrödinger operator with magnetic fields in domains with corners[END_REF]Prop. 11.4] and [START_REF] Bonnaillie-Noël | Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners[END_REF]Th. 7.1], we therefore conclude that

Q Ω,B [χ cor,s M φ] ≥ (µ 1 (α s )B -CB 1/2 ) χ cor,s M φ 2 .
(2.10) By a similar argument, we prove an analogous lower bound for the boundary contribution. Indeed, if B is large enough, the support of χ bd M is the union of N disjoint (c.f. the choice of L) domains localized near each piece of the smooth boundary and we rewrite

[χ bd M ] 2 = s∈Σ ([χ bd,s,- M ] 2 + [χ bd,s,+ M ] 2 ) with suppχ bd,s,± M ⊂ Ω ± s ∩ Ω, ∀s ∈ Σ.
Let s ∈ Σ. From the asymptotics of the ground state energy of [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]Thm. 11.1]) we get the following lower bound

H Ω ′ (B) for smooth domains Ω ′ ([
Q Ω,B [χ bd,s,± M φ] = Q Ω ± s ,B [χ bd,s,± M φ] ≥ Θ 0 B -2M 3 B 1/2 κ ± max (s) -C 0 (Ω ± s )B 1/3 χ bd,s,± M φ 2 , (2.11)
where M 3 is a universal constant, C 0 (Ω ± s ) is a domain-dependent constant and κ ± (s) denotes the maximal curvature of the boundary ∂Ω ± s . We can bound κ max (s) by

κ max := max s∈Σ,± κ ± max (s),
and similarly for C 0 (Ω ± s ). Then, there exists C independent of M and s such that

Q Ω,B [χ bd,s,± M φ] ≥ (Θ 0 B -CB 1/2 ) χ bd,s,± M φ 2 .
(2.12)

Using again the IMS-formula and (2.12), we can bound

Q Ω,B [χ bd M φ] ≥ s∈Σ,± Q Ω,B [χ bd,s,± M φ] -C B M 2 φ 2 ≥ (Θ 0 B -CB 1/2 ) χ bd M φ 2 -C B M 2 φ 2 .
(2.13)

We clearly get the result of Theorem 2.7 by combining (2.8) with (2.9), (2.10) and (2.13) and choosing M 0 , B 0 sufficiently large.

Using the lower bound (2.6) combined with the upper bound inherent in (2.4), one can get the following Agmon type estimate for the linear problem. Again we only state the result for completeness and without proof, since we will not use Theorem 2.8 in the remainder of the paper.

Theorem 2.8. Let ψ B be the ground state eigenfunction of H(B). Then there exist constants ǫ, C, B 0 > 0 such that

e ǫ √ Bdist(x,Σ) |ψ B (x)| 2 + B -1 |p BF ψ B (x)| 2 dx ≤ C ψ B 2 2 ,
for all B ≥ B 0 .

Basic estimates

We will need a number of standard results that we collect here for easy reference. First of all we have the usual L ∞ -bound for solutions to the Ginzburg-Landau equations (1.4), ψ ∞ ≤ 1.

(3.1)

The proof in [DGP] does not depend on regularity of the boundary, in particular, it is valid for domains with Lipschitz boundary.

The normalization of our functional E κ,H is such that E κ,H [0, F] = 0. So any minimizer (ψ, A) will have non-positive energy. Therefore, the only negative term, -κ 2 ψ 2 2 , in the functional has to control each of the positive terms. This leads to the following basic inequalities for minimizers,

p κHA ψ 2 ≤ κ ψ 2 , (3.2) H curl A -1 2 ≤ ψ 2 . (3.3) using (3.1), ψ 2 4 ≤ ψ 2 . (3.4)
The following lemma states that in two dimensions it is actually irrelevant whether we integrate the fields over Ω or over R 2 in the definition of E κ,H .

Lemma 3.1.

Let Ω be a bounded domain with Lipschitz boundary and let (ψ, A) be a (weak) solution to (1.4). Then curl (A -F) = 0 on the unbounded component of R 2 \ Ω.

Ω |A -F| 2 ≤ C 0 R 2 |curl A -1| 2 dx, (3.5) A -F 2 W 1,2 (Ω) ≤ C 0 R 2 |curl A -1| 2 dx. (3.6) Proof. Let b = curl (A -F). By Lemma 3.1, supp b ⊆ Ω. Define Γ 2 (x) = 1 2π log(|x|)
(the fundamental solution of the Laplacian in two dimensions), and w = Γ 2 * b. Then w ∈ Ḣ2 (R 2 ) and (see [GiTr])

∆w = b, ∇w L 2 (Ω) ≤ C(Ω) b L 2 (Ω) . (3.7) Let à = (-∂ 2 w, ∂ 1 w) ∈ Ḣ1 (R 2 ). Then div à = 0, curl à = b.
So we conclude that à = A -F, and therefore (3.5) follows from (3.7).

To establish (3.6) we use (3.5) together with the standard estimate

Da L 2 (R 2 ) ≤ C div a L 2 (R 2 ) + curl a L 2 (R 2 ) .
4. Non-linear Agmon estimates 4.1. Rough bounds on ψ 2 2 . In this chapter we prove that minimizers are localized near the boundary when H > κ. The precise meaning of that statement is given by Theorem 4.1 below. In since ψ ∞ ≤ 1, the L 2 -norm satisfies ψ 2 = o(1). We thus give a very precise and general upper bound to the field strength above which superconductivity is essentially a boundary phenomenon. Notice that this is the field which is usually called H C2 in the literature, although a precise mathematical definition is somewhat difficult to give.

The proof of Theorem 4.1 given below has been developed in cooperation with R. Frank.

Theorem 4.1 (Weak decay estimate).

Let Ω be a bounded domain with Lipschitz boundary. Then there exist C,

C ′ > 0, such that if (ψ, A) κ,H is a minimizer of E κ,H with κ(H -κ) ≥ 1/2, (4.1) then ψ 2 2 ≤ C { √ κ(H-κ) dist(x,∂Ω)≤1} |ψ(x)| 2 dx ≤ C ′ κ(H -κ) . (4.2)
Proof.

The last inequality is an easy consequence of (3.1), since there exists a constant C 1 > 0 (depending only on Ω) such that meas {x : dist(x, ∂Ω)

≤ λ} ≤ C 1 λ for all λ ∈ (0, 2]. Let χ ∈ C ∞ (R) be a standard non-decreasing cut-off function, χ = 1 on [1, ∞), χ = 0 on (-∞, 1/2).
Notice for later use that this implies that χ ′ ∞ ≥ 2. Let further λ > 0 (we will choose λ = 1/ κ(Hκ) at the end of the proof) and define χ λ : Ω → R by

χ λ (x) = χ(dist(x, ∂Ω)/λ).
Then χ λ is a Lipschitz function and supp χ λ ⊂ Ω. Combining the standard localization formula and (1.4a), we find

Ω |p κHA (χ λ ψ)| 2 dx - Ω |∇χ | 2 |ψ| 2 dx = ℜ χ 2 λ ψ, H κHA ψ = κ 2 |χ λ ψ| 2 dx -κ 2 χ 2 λ |ψ| 4 dx. (4.3)
Since χ λ ψ has compact support we have

Ω |p κHA (χ λ ψ)| 2 dx ≥ κH Ω (curl A)|χ λ ψ| 2 dx ≥ κH χ λ ψ 2 2 -κH curl A -1 2 χ λ ψ 2 4 . (4.4)
Using (3.4) and ( 3.3), we get from (4.3) and (4.4) that

κ(H -κ) χ λ ψ 2 2 ≤ κ ψ 2 χ λ ψ 2 4 -κ 2 χ 2 λ |ψ| 4 dx + χ ′ 2 ∞ λ -2 {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx ≤ 1 4 ψ 2 2 + χ ′ 2 ∞ λ -2 {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx + κ 2 (χ 4 λ -χ 2 λ )|ψ| 4 dx.
Notice that since χ ≤ 1, the last integral is negative and we thus find by dividing the integral

ψ 2 2 in two {κ(H -κ)-1/4} χ λ ψ 2 2 ≤ 1 4 (1 -χ 2 λ )|ψ| 2 dx + χ ′ 2 ∞ λ -2 {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx ≤ ( χ ′ 2 ∞ λ -2 + 1/4) {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx. Choose λ = 1/ κ(H -κ). By assumption κ(H -κ) -1/4 ≥ κ(H -κ)/2, and the conditions on χ, κ(H -κ) imply that χ ′ 2 ∞ λ -2 + 1/4 ≤ 2 χ ′ 2 ∞ λ -2 . Thus, χ λ ψ 2 2 ≤ 4 χ ′ 2 ∞ {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx. (4.5)
Consequently,

ψ 2 2 ≤ (4 χ ′ 2 ∞ + 1) {dist(x,∂Ω)≤λ} |ψ(x)| 2 dx. (4.6)
This finishes the proof of (4.2).

For stronger fields superconductivity is essentially localized to the corners.

Theorem 4.2 (Decay estimate on the boundary).

Suppose that Ω satisfies Assumption 1.3. For µ ∈ (Λ 1 , Θ 0 ), define

Σ ′ := {s ∈ Σ µ 1 (α s ) ≤ µ}, and b := inf s∈Σ\Σ ′ {µ 1 (α s ) -µ}.
(4.7)

(in the case Σ = Σ ′ , we set b := Θ 0 -µ). There exist κ 0 , C, C ′ , M > 0, such that if (ψ, A) κ,H is a minimizer of E κ,H with H κ ≥ µ -1 , κ ≥ κ 0 , (4.8) then ψ 2 2 ≤ C {κ dist(x,Σ ′ )≤M } |ψ(x)| 2 dx ≤ C ′ κ 2 . (4.9)
Proof.

To prove this result, we follow the same procedure as in the proof of Theorem 4.1. Let δ = b/2, and let M 0 = M 0 (δ) be the constant from Theorem 2.7. Let χ ∈ C ∞ (R) be a standard non-decreasing cut-off function,

χ = 1 on [1, ∞), χ = 0 on (-∞, 1/2),
and

let λ = 2M 0 / √ κH. Define χ λ : Ω → R, by χ λ (x) = χ(dist(x, Σ ′ )/λ).
Then χ λ is a Lipschitz function and supp χ λ ∩ Σ ′ = ∅. Combining the standard localization formula and (1.4a), we find as previously

Ω |p κHA (χ λ ψ)| 2 dx - Ω |∇χ λ | 2 |ψ| 2 dx = ℜ χ 2 λ ψ, H κHA ψ ≤ κ 2 χ λ ψ 2 2 . (4.10)
As in (4.4), we need a lower bound to Ω |p κHA (χ λ ψ)| 2 dx. Since supp χ λ ∩ ∂Ω = ∅, we cannot argue as in (4.4). Therefore, we will introduce the constant magnetic field F for which we have such an estimate, namely Theorem 2.7. We can write

Ω |p κHA (χ λ ψ)| 2 dx ≥ (1 -ε) Ω |p κHF (χ λ ψ)| 2 dx -ε -1 Ω (κH) 2 |F -A| 2 (χ λ ψ)| 2 dx. (4.11)
Theorem 2.7 and the choice of λ imply that

Ω |p κHF (χ λ ψ)| 2 dx ≥ inf s∈Σ\Σ ′ µ 1 (α s ) -δ κH χ λ ψ 2 2 = µ + b 2 κH χ λ ψ 2 2 .
(4.12)

We now have to give a lower bound to the second part of the right side of (4.11). We can estimate

Ω (κH) 2 |F -A| 2 |χ λ ψ| 2 dx ≤ (κH) 2 A -F 2 4 χ λ ψ 2 4 . (4.13)
By Sobolev inequalities, (3.6) and (3.3), we deduce

(κH) 2 F -A 2 4 ≤ Cκ 2 H 2 F -A 2 W 1,2 (Ω) ≤ Cκ 2 H 2 curlA -1 2 L 2 (R 2 ) ≤ Cκ 2 ψ 2 2 . (4.14)
Let us now estimate χ λ ψ 2 4 . According to (3.1) and the property of the cut-off function 0 ≤ χ λ ≤ 1, we can bound |χ λ ψ| from above by 1 and deduce, using also Theorem 4.1,

χ λ ψ 2 4 = Ω |χ λ ψ| 4 dx ≤ Ω |χ λ ψ| 2 dx ≤ C √ κ . (4.15)
Inserting (4.12), (4.13), (4.14) and (4.15) in (4.11), we obtain

Ω |p κHA (χ λ ψ)| 2 dx ≥ (1 -ε) µ + b 2 κH χ λ ψ 2 2 -Cε -1 κ 3/2 ψ 2 2 . (4.16)
We insert (4.16) in (4.10). Then

(1 -ε) µ + b 2 κH -κ 2 -Cε -1 κ 3/2 {dist(x,Σ ′ )≥λ} |ψ| 2 dx ≤ (Cε -1 κ 3/2 + χ ′ 2 ∞ λ -2 ) {dist(x,Σ ′ )≤λ} |ψ| 2 dx, (4.17)
Assumption (4.8) leads to the lower bound

(1 -ε) µ + b 2 κH -κ 2 -Cε -1 κ 3/2 ≥ b 4 κH, (4.18)
as soon ε is small enough and κ large enough.

Once ε is fixed and with λ = 2M 0 / √ κH, we find

Cε -1 κ 3/2 + χ ′ ∞ λ -2 ≤ cκH. (4.19)
Combining (4.17), (4.18) and (4.19), we deduce

{dist(x,Σ ′ )≥λ} |ψ| 2 dx ≤ C {dist(x,Σ ′ )≤λ} |ψ| 2 dx. (4.20)
It follows easily that

ψ 2 2 ≤ (C + 1) {dist(x,Σ ′ )≤λ} |ψ| 2 dx.
Inserting the choice λ = 2M 0 / √ κH and the condition (4.8) on H, this clearly implies (4.9).

Exponential localization.

In order to obtain exponential decay in the interior of the domain, we need the following energy estimate, Lemma 4.3, for functions located away from the boundary.

Lemma 4.3.

Let Ω ⊂ R 2 be a bounded domain with Lipschitz boundary. There exist constants

C 0 , C 1 > 0 such that if κ(H -κ) ≥ C 0 and (ψ, A) is a minimizer of E κ,H , then for all φ ∈ C ∞ 0 (Ω) we have (-i∇ -κHA)φ 2 2 ≥ κH 1 -C 1 ψ 2 φ 2 2 .
In particular, using the estimate on ψ 2 from Theorem 4.1 we find

(-i∇ -κHA)φ 2 2 ≥ κH 1 - C ′ 1 4 κ(H -κ) φ 2 2 .
Proof.

We estimate, for φ ∈ C ∞ 0 (Ω),

(-i∇ -κHA)φ 2 2 ≥ κH Ω curl A|φ| 2 dx ≥ κH φ 2 2 -κH curl A -1 2 φ 2 4 . (4.21)
By the Sobolev inequality, for φ ∈ C ∞ 0 (R 2 ), and scaling we get, for all η > 0 and with a universal constant C Sob , the estimate

φ 2 4 ≤ C Sob η ∇|φ| 2 2 + η -1 φ 2 2 . (4.22)
We can estimate ∇|φ| 2 2 by (-i∇ -κHA)φ 2 2 by the diamagnetic inequality. Choosing, η = η ′ C Sob κH curl A-1 2 , for some η ′ > 0, we thus find, using (3.3), (4.21) and (4.22),

(-i∇ -κHA)φ 2 2 ≥ κH φ 2 2 -η ′ (-i∇ -κHA)φ 2 2 -(η ′ ) -1 C 2 Sob (κH) 2 curl A -1 2 2 φ 2 2 ≥ κH φ 2 2 1 -(η ′ ) -1 C 2 Sob κ H ψ 2 2 -η ′ (-i∇ -κHA)φ 2 2 . (4.23)
By assumption κ/H ≤ 1. We take η ′ = ψ 2 and find

(1 + ψ 2 ) (-i∇ -κHA)φ 2 2 ≥ κH(1 -C 2 Sob ψ 2 ) φ 2 2 . (4.24)
By Theorem 4.1 we have

1 -C 2 Sob ψ 2 1 + ψ 2 ≥ 1 -2C 2 Sob ψ 2 ,
if κ(Hκ) is sufficiently big. This finishes the proof of Lemma 4.3.

By standard arguments Lemma 4.3 implies Agmon estimates in the interior.

Theorem 4.4 (Normal Agmon estimates).

Let Ω be a bounded domain with Lipschitz boundary and let b > 0. There exist

M, C, ǫ, κ 0 > 0, such that if (ψ, A) is a minimizer of E κ,H with H κ ≥ 1 + b, κ ≥ κ 0 , then Ω e 2ǫ √ κHt(x) |ψ| 2 + 1 κH (-i∇ -κHA)ψ 2 dx ≤ C {t(x)≤ M √ κH } |ψ| 2 dx. (4.25)
Here t(x) := dist(x, ∂Ω).

Proof.

The function t(x) = dist(x, ∂Ω) defines a Lipschitz continuous function on Ω. In particular, ∇t ∈ L ∞ (Ω). Let χ ∈ C ∞ (R) be a non-decreasing function satisfying

χ = 1 on [1, ∞), χ = 0 on [-∞, 1/2).
Define the (Lipschitz continuous) function χ M on Ω by χ M (x) = χ( t(x) √ κH M

). We calculate, using (1.4a) and the IMS-formula

κ 2 exp ǫ √ κHt χ M ψ 2 2 ≥ ℜ exp 2ǫ √ κHt χ 2 M ψ, κ 2 (1 -|ψ| 2 )ψ = Ω p κHA e ǫ √ κHt χ M ψ 2 dx - Ω ∇(e ǫ √ κHt χ M )ψ 2 dx. (4.26)
Combining Theorem 4.1 with Lemma 4.3 there exists g with g = o(1) at ∞, such that

Ω p κHA e ǫ √ κHt χ M ψ 2 dx ≥ κH(1 + g(κH)) e ǫ √ κHt χ M ψ 2 2 .
Since H κ ≥ 1 + b, we therefore find, with some constant C independent of κ, H, ǫ and M

1 + g(κH) - 1 1 + b e ǫ κHt χ M ψ 2 2 ≤ Cǫ 2 ∇t 2 ∞ e ǫ √ κHt χ M ψ 2 2 + C ∇t 2 ∞ M 2 Ω e 2ǫ √ κHt(x) χ ′ t(x) √ κH M ψ(x) 2 dx.
(4.27)

For κ sufficiently big we have, since

H ≥ (1 + b)κ, 1 + g(κH) - 1 1 + b ≥ b/2.
We choose ǫ sufficiently small that Cǫ 2 ∇t 2 ∞ < b/4 and finally obtain for some new constant

C ′ e ǫ √ κHt χ M ψ 2 2 ≤ C ′ e 2ǫM M 2 { √ κHt(x)≤M } |ψ(x)| 2 dx. (4.28)
On the support of 1χ M the exponential e ǫ √ κHt is bounded, so we see that

e ǫ √ κHt ψ 2 2 ≤ C ′′ { √ κHt(x)≤M } |ψ(x)| 2 dx, (4.29)
which is part of the estimate (4.25).

It remains to estimate the term with (-i∇ -κHA)ψ in (4.25). This follows from the same considerations upon inserting the bound (4.29).

Lemma 4.5. Suppose that Ω ⊂ R 2 satisfies Assumption 1.3. For µ ∈ (Λ 1 , Θ 0 ), define

Σ ′ := {s ∈ Σ µ 1 (α s ) ≤ µ},
and b := inf s∈Σ\Σ ′ {µ 1 (α s ) -µ}.

(4.30)

(in the case Σ = Σ ′ , we set b := Θ 0 -µ). There exist M 0 > 0 such that if (ψ, A) is a minimizer of E κ,H , then for all φ ∈ C ∞ (Ω) such that dist(supp φ, Σ ′ ) ≥ M 0 / √ κH, we have (-i∇ -κHA)φ 2 L 2 (Ω) ≥ µκH 1 + b 4 φ 2 L 2 (Ω) , (4.31)
for κH sufficiently large.

Proof.

Let δ = b/2 and let M 0 = M 0 (δ) be the constant from Theorem 2.7. We estimate, for φ ∈ C ∞ (Ω) such that dist(supp φ, Σ ′ ) ≥ M 0 / √ κH,

(-i∇ -κHA)φ 2 2 = (-i∇ -κHF)φ + κH(F -A)φ 2 2 ≥ (1 -ε) Ω |(-i∇ -κHF)φ| 2 dx -ε -1 Ω (κH) 2 |F -A| 2 |φ| 2 dx. (4.32)
Using Theorem 2.7 and the support properties of φ, we have

Ω |(-i∇ -κHF)φ| 2 dx ≥ inf s∈Σ\Σ ′ µ 1 (α s ) -δ κH φ 2 2 = µ + b 2 κH φ 2 2 . (4.33)
Using the Cauchy-Schwarz inequality, (4.14) and Theorem 4.2, we can bound the last term of (4.32). We choose η = C Sob ε 2 κH , then (4.36) becomes

Ω (κH) 2 |F -A| 2 |φ| 2 dx ≤ (κH) 2 A -F 2 4 φ 2 4 ≤ Cκ 2 ψ 2 2 φ 2 4 ≤ C |φ| 2 4 . ( 4
1 + C 2 Sob ε 3 κH (-i∇ -κHA)φ 2 2 ≥ κH (1 -ε) µ + b 2 -ε φ 2 2 .
(4.37)

If we choose ε sufficiently small and independent of κ, H (actually, since µ+b/2 ≤ 1, ε = b/8 will do) then (4.31) follows.

By standard arguments Lemma 4.5 implies the Agmon estimates given in Theorem 1.6. Proof of Theorem 1.6. The function t ′ (x) := dist(x, Σ ′ ) defines a Lipschitz continuous function on Ω. In particular, |∇t ′ | ≤ 1. Let χ ∈ C ∞ (R) be a non-decreasing function satisfying

χ = 1 on [1, ∞), χ = 0 on [-∞, 1/2). Define the function χ M on Ω by χ M (x) = χ( t ′ (x) √ κH M
). Using Lemma 4.5 there exists β > 0, such that if M, κH are sufficiently large, then

Ω p κHA e ǫ √ κHt ′ χ M ψ 2 dx ≥ µκH(1 + β) e ǫ √ κHt ′ χ M ψ 2 2 .
Using (4.26) and the assumption H κ ≥ µ -1 , there exists some constant C independent of κ, H, ǫ and M such that

βµ e ǫ √ κHt ′ χ M ψ 2 2 ≤ Cǫ 2 ∇t ′ 2 ∞ e ǫ √ κHt ′ χ M ψ 2 2 (4.38) + C ∇t ′ 2 ∞ M 2 Ω e 2ǫ √ κHt ′ (x) χ ′ t ′ (x) √ κH M ψ(x) 2 dx.
We achieve the proof of Theorem 1.6 with arguments similar to the ones of the proof of Theorem 4.4.

Proof of Theorem 1.4

Combining Proposition 2.4 and Lemma 2.5 it only remains to prove (1.9). We will prove that for large κ the following two statements are equivalent.

(1) There exists a minimizer (ψ, A) of E κ,H with ψ 2 = 0.

(2) The parameters κ, H satisfy κ 2λ 1 (κH) > 0.

(5.1) Suppose first that (5.1) is satisfied. Let u 1 (κH) be the normalized ground state eigenfunction of H(κH) and let t > 0. Then, for t 2 < 2 κ 2 -λ1(κH)

κ 2 u1(κH) 4 4 , E κ,H [tu 1 (κH), F] = t 2 [λ 1 (κH) -κ 2 ] + κ 2 2 t 4 u 1 (κH) 4 4 < 0. (5.2)
This shows that (2) implies (1). Notice that this first part did not need the assumption that κ is large. However, for large κ we know that (5.1) is satisfied iff H < H lin C3 (κ) (defined in Lemma 2.5). Suppose that (ψ, A) is a non-trivial minimizer of E κ,H . We may assume that H > (1 + b)κ for some b > 0, because by Proposition 2.4, (5.1) is satisfied for κ ≥ κ 0 , H < H lin C3 (κ), where H lin C3 (κ) has the asymptotics given in Lemma 2.5. Furthermore, we may assume that H ≤ T κ for some T > 0. This follows from [GiPh]-we give the details for completeness: Since ψ = 0, we have

0 < λ 1 (κH) ψ 2 2 ≤ Ω |p κHF ψ| 2 dx ≤ 2 Ω |p κHA ψ| 2 dx + 2(κH) 2 Ω |A -F| 2 |ψ| 2 dx.
We now use, (3.1) and Lemma 3.2 to obtain

0 < λ 1 (κH) ψ 2 2 ≤ C Ω |p κHA ψ| 2 dx + (κH) 2 R 2 |curl A -1| 2 dx ≤ Cκ 2 ψ 2 2
, where the last inequality holds since E κ,H [ψ, A] ≤ 0. Since λ 1 (B) increases linearly in B we deduce that H = O(κ).

From the discussion above, we know that we may assume

(1 + b)κ ≤ H ≤ b -1 κ,
for some b > 0. By Theorem 4.1 we therefore find, for some C > 0,

ψ 2 2 ≤ C {dist(x,∂Ω)≤ 1 κ } dx 1/2 ψ 2 4 ≤ C ′ ψ 2 4 √ κ .
(5.3)

Finally, ψ 0 satisfies the uniform bound,

ψ 0 ∞ ≤ µ 1 µ 2 .
One easily verifies the following scaling property.

Proposition 6.2.

Let Λ > 0. Then the functional,

ψ → Γα |(-i∇ -Λ -2 F)ψ| 2 -µ 1 Λ -2 |ψ| 2 + µ 2 2 Λ -2 |ψ| 4 dx, defined on {ψ ∈ L 2 (Γ α ) (-i∇ -Λ -2 F)ψ ∈ L 2 (Γ α )} is minimized by ψ0 (y) = ψ 0 (y/Λ)
, where ψ 0 is the minimizer of J α µ1,µ2 . In particular,

inf ψ Γα |(-i∇ -Λ -2 F)ψ| 2 -µ 1 Λ -2 |ψ| 2 + µ 2 2 Λ -2 |ψ| 4 dx = E α µ1,µ2
.

By continuity of E α µ1,µ2 we get the following consequence.

Proposition 6.3. Suppose that κ H(κ) → µ < Θ 0 as κ → ∞, and that d 1 (κ), d 2 (κ) → 1 as κ → ∞. Then the ground state energy of the functional

ψ → Γα |(-i∇ -κHF)ψ| 2 -d 1 (κ)κ 2 |ψ| 2 + d 2 (κ) κ 2 2 |ψ| 4 dx,
tends to E α µ,µ as κ → ∞. 6.2. Coordinate changes. Let s ∈ Σ. By the assumption that ∂Ω is a curvilinear domain there exists r s > 0 and a local diffeomorphism Φ s of R 2 such that Φ s (s) = 0, (DΦ s )(s) ∈ SO(2) and Φ s B(s, r s ) ∩ Ω = Γ αs ∩ Φ s (B(s, r s )).

Let u, A = (A 1 , A 2 ) ∈ C ∞ 0 (B(s, r s )) and define ũ(y) = u(Φ -1 s (y)). Let furthermore, B(y) = B(Φ -1 s (y)), where B(x) = curl A. Then the quadratic form transforms as

Ω |(-i∇ -A)u(x)| 2 dx = Γα s (-i∇ -Ã)ũ(y), G(y)(-i∇ -Ã)ũ(y) | det DΦ -1 s (y)| dy. (6.2)
Here G(y) = (DΦ s )(DΦ s ) T x=Φs(y) , and à = ( Ã1 , Ã2 ) satisfies (6.4) which is the 'easy' part of (1.12).

A 1 dx 1 + A 2 dx 2 = Ã1 dy 1 + Ã2 dy 2 , so
The inequality (6.4) follows from a calculation with an explicit trial state. The test functions will be of the form A = F and ψ(x) = s∈Σ ψ s (Φ s (x)), with ψ s (y) = e iκHηs ψ αs 1,1 ( √ κHy)χ(|y|).

Here η s ∈ C ∞ (R 2 , R) is a gauge function, χ is a standard cut-off function, χ = 1 on a neighborhood of 0, supp χ ⊂ B(0, r), with r = min s∈Σ {r s }, and ψ αs 1,1 is the minimizer of J αs 1,1 . The proof of (6.4) is a straight forward calculation similar to the lower bound (given below) and will be omitted. Notice though that the decay estimates (6.1) for the minimizers ψ αs 1,1 imply that ψ αs 1,1 ( √ κHy)χ(|y|)-ψ αs 1,1 ( √ κHy), is exponentially small. Lower bounds Let (ψ, A) be a minimizer of E κ,H . Define χ 1 , χ 2 ∈ C ∞ (R) to be a standard partition of unity, χ 1 is non-increasing, χ 2 1 + χ 2 2 = 1, χ 1 (t) = 1 for t ≤ 1, χ 1 (t) = 0 for t ≥ 2.

For s ∈ Σ, let φ s (x) = χ 1 κ 1-ǫ dist(x, s)

with ǫ > 0, and define φ 0 = 1 -s∈Σ φ 2 s . Notice that when κ is sufficiently large and s, s ′ ∈ Σ, s = s ′ , then φ s φ s ′ = 0. Therefore, using the Agmon estimates, the IMS-localization formula and the estimate ψ ∞ ≤ 1, we can write, We choose 0 < ǫ < 1/3 arbitrary. Using Proposition 6.3 and combing (6.5) and (6.11) we find the lower bound inherent in (6.4), i.e. This finishes the proof of Theorem 1.7.

Figure 1 .

 1 Figure 1. µ 1 (α) vs. α/π for α ∈ [0, 1.25π].

Figure 2 .Figure 3 .

 23 Figure 2. Definition of Ω s

∂

  y1 Ã2 -∂ y2 Ã1 = | det DΦ -1 s (y)| B(y).(6.3) 6.3. Proof of Theorem 1.7.Upper boundsWe indicate here how to obtain the inequality inf(ψ,A) E κ,H(κ) [ψ, A] ≤ s∈Σ E αs µ,µ + o(1),

E| 2 -

 2 κ,H [ψ, A] ≥ s∈Σ E κ,H [φ s ψ, A] + O(κ -∞ ).(6.5)By the Sobolev imbedding W 1,2 (Ω) → L 4 (Ω), Lemma 3.2 combined with (3.3), and the Agmon estimate we get(κH) 2 A -F 2 4 ≤ C(κH) 2 A -F 2 W 1,2 (Ω) ≤ C ′ (κH) 2 curl A --κHA)(φ s ψ) 2 dx ≥ (1κ -1/2 ) Ω (-i∇ -κHF)(φ s ψ) 2 dxκ 1/2 (κH) 2 A -≤ C {dist(x,Σ)≤M κ -1 } 1 dx ≤ C ′ κ -1 .Now consider the change of coordinates Φ s from subsection 6.2. For sufficiently large values of κ we have suppφ s ⊂ B(s, r s ). Define ψs = (φ s ψ) • Φ -1 s . Since | det DΦ s (0)| = 1, we get by Taylor's formula that | det DΦ s | -1 ≤ Cκ -1+ǫ , on supp ψs .Consider the transformed magnetic field as in (6.3). We define β(y) := | det DΦ -1 s (y)| B(y) = | det DΦ -1 s (y)| = 1 + O(κ -1+ǫ), (6.8) on supp ψs . We look for à = ( Ã1 , Ã2 ) such that ∂ y1 Ã2 -∂ y2 Ã1 = β(y). One choice of a solution isà = -y 2 /2, à -F L ∞ (B(0,Cκ -1+ǫ )) ≤ C ′ κ -2+2ǫ . Thus (κH) 2 | à -F| 2 | ψs | 2 dy ≤ C(κH) 2 κ -4+4ǫ | ψs | 2 dy ≤ C ′ κ -2+6ǫ . (6.9) Therefore, for some η ∈ C ∞ (Ω, R) we find Ω (-i∇ -κHF)(φ s ψ) 2 dx = Γα s (-i∇ -κH Ã)(e iκHη ψs ), G(y)(-i∇ -κH Ã)(e iκHη ψs ) | det DΦ -1 s | dy ≥ (1 Cκ -1+ǫ ) Γα s (-i∇ -κH Ã)(e iκHη ψs ) 2 dy ≥ -Cκ -1+ǫ ) (1κ -1+3ǫ ) à -F| 2 | ψs | 2 dy ≥ (1 -2Cκ -1+3ǫ ) Γα s (-i∇ -κHF)(e iκHη ψs ) 2 dy + O(κ -1+3ǫ). (6.10) By (6.10) we findE κ,H [φ s ψ, A] ≥ (1 -C 1 κ -1+3ǫ )Γα s (-i∇ -κHF)(e iκHη ψs ) (1 + C 2 κ -1+3ǫ ) κ 2 2 |e iκHη ψs | 2 + κ 4 |e iκHη ψs | 4 dy + O(κ -1+3ǫ ). (6.11)

Acknowledgements

It is a pleasure to acknowledge discussions on this and related subjects with X. Pan and R. Frank. Furthermore, without the discussions with and encouragement from B. Helffer this work would never have been carried through.

Both authors were supported by the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). The first author is partly supportted by the ANR project 'Macadam' JCJC06-139561. The second author is supported by a Skou-stipend from the Danish Research Council and has also benefitted from support from the European Research Network 'Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems' with contract number HPRN-CT-2002-00277. Furthermore, the second author wants to thank CIMAT in Guanajuato, Mexico for hospitality.

Proof.

The second equation, (1.4b) reads in the exterior of Ω, using that curl F = 1,

Thus we see that curl (A-F) is constant on each connected component of R 2 \Ω and since it has to be in L 2 it must therefore vanish on the unbounded component.

Lemma 3.2.

There exists a constant C 0 (depending only on Ω) such that if (ψ, A) is a (weak) solution of the Ginzburg-Landau equations (1.4), then Since (ψ, A) is a non-trivial minimizer, E κ,H [ψ, A] ≤ 0. So we also have

(5.4)

The inequality (5.3) therefore becomes,

(5.5)

By Cauchy-Schwarz we can estimate

(5.6)

So we find, by inserting (5.5), (5.4) and using Cauchy-Schwarz,

(5.7)

Since E κ,H [ψ, A] ≤ 0, we get using Lemma 3.2 and a Sobolev imbedding,

, which permits to conclude that (5.1) is satisfied.

Thus (1) and ( 2) are equivalent for large κ which implies (1.9). This finishes the proof of Theorem 1.4.

Energy of minimizers

Basic properties.

In the case where H κ → 1 µ , with Λ 1 = min s∈Σ µ 1 (α s ) < µ < Θ 0 , superconductivity is dominated by the corners. The asymptotics of the ground state energy in this case is given by Theorem 1.7 which we will prove in the present section.

Recall the functionals J α µ1,µ2 with ground state energy E α µ1,µ2 defined on angular sectors Γ α by (1.11). We give the following proposition without proof, since it is completely analogous to the similar statements for E κ,H . Proposition 6.1. The map (0, Θ

µ1,µ2 = 0 and ψ = 0 is a minimizer. If µ 1 > µ 1 (α), there exists a non-trivial minimizer ψ 0 of J α µ1,µ2 . Furthermore, there exist constants a, C > 0 such that Γα e 2a|x| |ψ 0 (x)| 2 + |(-i∇ -F)ψ 0 | 2 dx ≤ C.

(6.1)


Oleg Szehr
email: oleg.szehr@posteo.de

research documents, whether they are published or not. The documents may come

On the asymptotic behavior of Jacobi polynomials with varying parameters

ON THE ASYMPTOTIC BEHAVIOR OF JACOBI POLYNOMIALS WITH VARYING PARAMETERS

OLEG SZEHR AND RACHID ZAROUF

Abstract. We study the large n behavior of Jacobi polynomials with varying parameters P (an+α,β) n

(1 -2λ 2 ) for a > -1 and λ ∈ (0, 1). This appears to be a well-studied topic in the literature but some of the published results are unnecessarily complicated or incorrect. The purpose of this paper is to provide a simple and clear discussion and to highlight some flaws in the existing literature. Our approach is based on a new representation for P (an+α,β) n

(1-2λ 2 ) in terms of two integrals. The integrals' asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. In summary we prove that if a ∈ ( 2λ 1-λ , ∞) then λ an P (an+α,β) n

(1 -2λ 2 ) shows exponential decay and we derive exponential upper bounds in this region. If a ∈ ( -2λ 1+λ , 2λ 1-λ ) then the decay of λ an P (an+α,β) n

(1 -2λ 2 ) is O(n -1/2
) and if a ∈ { -2λ 1+λ , 2λ 1-λ } then λ an P (an+α,β) n

(1 -2λ 2 ) decays as O(n -1/3 ). Lastly we find that if a ∈ (-1, -2λ 1+λ ) then λ an P

Introduction and Summary

Jacobi polynomials P (α, β) n constitute a class of orthogonal degree-n polynomials, which depend on two parameters α, β ∈ R. An explicit representation is [START_REF] Szegö | Orthogonal polynomials[END_REF](4.3.2) p. 68]

P (α, β) n (x) = n µ=0 n + α n -µ n + β µ x -1 2 µ x + 1 2 n-µ
.

Historically Jacobi polynomials have experienced exhaustive research, see for example [WW, GZ] for an overview of existing results. In this paper we are concerned with with the behavior of Jacobi polynomials for large n, which is a classical topic in asymptotic analysis.

To gain some background we state the probably most famous result: Darboux's asymptotic formula, which we cite in the formulation of Szegö [START_REF] Szegö | Orthogonal polynomials[END_REF]Theorem 8.21.8,p. 196].

Proposition 1. Let α, β ∈ R and let θ ∈ [ , π -] then the Jacobi polynomials satisfy the following asymptotic expansion

P (α,β) n (cos(θ)) = 1 √ nπ k(θ) cos(N θ + γ) + O 1 n 3/2 , N = n + (α + β + 1)/2, γ = -(α + 1 2 )π/2, 0 < θ < π.
The main assertion of Darboux's formula is that for fixed α, β Jacobi polynomials decay for large n in first order like O(n -1/2 ). The asymptotic growth of Jacobi polynomials with varying parameters, λ an P (α+an,β) n

(1 -2λ 2 ) where -1 < α, β, a, is studied in multiple articles, see e.g. [BG, CI, FFN, GS, SI, SV]. The behavior is non-trivial in that within the a-range (-1, ∞) different types of asymptotic bounds occur. For instance, it is the main result of [BG, GS, CI, SI] that for a ∈ (-2λ 1+λ , 2λ 1-λ ) in first order a O(n -1/2 ) asymptotic behavior can be observed but the explicit formulas are inconsistent. The main point of our article is to tidy up the inconsistent and partially incorrect findings in the literature and provide a simple access to the topic. We begin by stating our main result and relating it to the existing literature.

Theorem 2. Let α, β > -1, a > -1 and λ ∈ (0, 1). We have the following asymptotic expansion as n → ∞:

(1) If a ∈ (-2λ 1+λ , 2λ 1-λ ) then

P (α+an, β) n (1 -2λ 2 ) = 2 nπ λ -α-an ((1 -λ 2 )(a + 1)) -β 2 ((1 -λ 2 )((a + 2)λ + a)((a + 2)λ -a))
1 4

• cos (n + 1)h(ϕ

+ ) + (α -a)ϕ + + (β -1)ψ+ π 4 1 + O(n -1
) .

The phases ϕ + , h(ϕ + ), ψ ∈ [0, 2π] depend on a, λ and are given explicitly in Proposition 4 below.

(2) If a = 2λ 1-λ then

P (an+α, β) n (1 -2λ 2 ) = λ -an-α (1 + λ) -β 3 2/3 Γ(2/3)n 1/3 λ 1/3 (1 + λ) 1/3 1 + O(n -1/3 ) . If a = -2λ 1+λ then P (an+α, β) n (1 -2λ 2 ) = Γ(1/3)λ -α-an (1 -λ) -β 3 2/3 πn 1/3 λ 1/3 (1 -λ) 1/3 • √ 3 2 cos ((an + α)π) - √ 3 sin ((an + α)π) 1 + O(n -1/3 ) . (3) If a ∈ (-1, -2λ 1+λ ) ∪ ( 2λ 1-λ , ∞) and an + α is integer then the quantity λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 )
decays exponentially with n, see Proposition 7 for details. (4) If a ∈ (-1, -2λ 1+λ ) and an + α is not integer then the quantity

λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 )
increases exponentially with n, see Proposition 7 for details. If a ∈ ( 2λ 1-λ , ∞) and an + α is not an integer then the above quantity decays exponentially, see Proposition 7 for details.

The same topic was studied in earlier publications. Chen, Ismail and Izen [CI, SI] derive asymptotic bounds as in Theorem 2 but their findings are inaccurate in the whole parameter range of a. [GS, FFN] derive (among other things) the asymptotic behavior in the range a ∈ (-2λ 1+λ , 2λ 1-λ ) ∪ ( 2λ 1-λ , ∞) and in this range we recover their findings using our new method. In the range a ∈ (-1, -2λ

1+λ ) [SI] claim exponential decay for λ an P (α+an,β) n

(1 -2λ 2 ) but we find that this happens iff an + α is integer. In other words exponential decay occurs along a subsequence of values n, where an+α is integer. If an+α is not integer we prove that λ an P (α+an,β) n

(1 -2λ 2 ) is increasing exponentially. Finally we derive the asymptotic behavior at the saddle points a ∈ { -2λ 1+λ , 2λ 1-λ }, where the asymptotics were not studied before.

Various methods are employed in the cited publications to access the problem. Gawronksi-Shawyer [GS] as well as Saff-Varga [SV] rely on the method of steepest descent [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF]p. 147] [FO0,p. 136], while Chen-Ismail [CI] and Izen [SI] make use of Darboux's asymptotic method and generating functions [START_REF] Olver | Asymptotics and Special Functions[END_REF][START_REF] Szegö | Orthogonal polynomials[END_REF]. In this article we derive a simple integral representation for the Jacobi polynomials with varying parameters, see Lemma 3. The lemma provides a representation of λ an P (α+an,β) n

(1 -2λ 2 ) as a sum of two integrals, which can be treated using standard tools from asymptotic analysis:

(1) The first integral is a so-called generalized Fourier integral [AE], [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF]Chapter 6.5], which is of the form ˆb a g(t)e inh(t) dt, with continuous real functions g, h. To determine the leading order behavior we employ the method of stationary phase, which is commonly used to study asymptotic properties of oscillatory integrals, see Section 3.1. The method is similar to Laplace's method (see below) in that the leading order contribution comes from a small neighborhood around the stationary points of h. Our analysis relies on the well-known fact [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] [BO, Chapter 6.5] that if ξ is a stationary point of h such that g(ξ) = 0 then (a) the integral goes to 0 as

n -1/2 if h (ξ) = 0, (b) the integral goes to 0 as n -1/3 if h (c) = 0 but h (3) (c) = 0.
(2) The second integral contributes iff α+an is not an integer and is a so-called Laplace type integral [START_REF] Olver | Asymptotics and Special Functions[END_REF]Chapter 3], which is of the form ˆb a g(t)e nh(t) dt, with continuous, real functions g, h. We use Laplace's method to determine the asymptotic behavior, see Section 3.2. The idea is that if the real continuous function h has a maximum at a point ξ ∈ [a, b] and if g(ξ) = 0 then as n grows only values in an immediate neighborhood of ξ should contribute to the integral, see [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF][START_REF] Olver | Asymptotics and Special Functions[END_REF].

Integral representation for Jacobi Polynomials

The main point in our approach is to find a good integral representation for Jacobi Polynomials. The emerging integrals are treated using standard methods from asymptotic analysis [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF][START_REF] Olver | Asymptotics and Special Functions[END_REF][START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]. Although simple, we regard the below lemma as the key technical innovation of our work.

Lemma 3. [Integral representation for Jacobi polynomials] Let n be an integer, a > -1, α, β > -1, λ ∈ (0, 1). For any x ∈ (λ, 1/λ) we have the following integral representation for the Jacobi polynomials with varying parameters

λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 ) = 1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=xe iϕ dϕ - sin (π(α + an)) π ˆx 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt,
where (•) refers to the real part of a complex number.

This representation is particularly suited for asymptotic analysis. The first integral is a generalized Fourier integral and its large n behavior is derived using the method of stationary phase. The second one is a Laplace integral and can be studied using Laplace's asymptotic results. Curiously the second integral contributes iff an + α is not an integer. As a consequence it is necessary to consider two different types of subsequences of values n in the study of the asymptotic behavior of λ an+α (1 -λ 2 ) β P (an+α, β) n

(1 -2λ 2 ). For a subsequence of n, where an + α is integer the second integral does not contribute but if an + α is not an integer the second integral contributes and changes the asymptotic behavior of λ an+α (1 -λ 2 ) β P (an+α, β) n

(1 -2λ 2 ) "discontinuously". A similar phenomenon was observed in [KM] for a ∈ (-1, 0) and β = bn with b ∈ (-1, 0) and a + b < -1.

In particular it suggests that the limiting behavior of zeros of the corresponding Jacobi polynomials [MMO] will be very sensitive to the proximity of an, bn, an + bn to integers [KM].

Proof. We investigate the complex function

f : z → z (a+1)n+α (1 -λz) n+β (z -λ) n+1
, where powers are taken with respect to the principal branch. We integrate f along a closed contour γ in the complex plane, see Figure 2

.1. The contour is composed of four curves γ = γ 1 ⊕ γ 2 ⊕ γ 3 ⊕ γ 4 with x ∈ (λ, 1/λ) and γ 1 : ϕ → xe iϕ , ϕ ∈ (-π, π] γ 2 : ϕ → e -iϕ , ϕ ∈ (-π, π] γ 3 : t → t + i˜ , t ∈ [-x, 0] γ 4 : t → -t -i˜ , t ∈ [0, x]. γ 2 γ 1 γ 3 γ 4 γ 5 λ 1 Figure 2.1. Contours γ = γ 1 ⊕ γ 2 ⊕ γ 3 ⊕ γ 4 and γ 5 for x = 1
Due to holomorphy we have that

1 2πi ˆγ1 ⊕γ 2 ⊕γ 3 ⊕γ 4 f dz = 1 2πi ˆγ5 f dz,
where γ 5 denotes a small circle around the pole λ, see Figure 2.1,

γ 5 : ϕ → λ + se iϕ , ϕ ∈ (-π, π].
The lemma is proved by computing each of the five integrals individually.

• We first prove that

1 2πi ˆγ1 f dz = 1 2π ˆπ -π z (a+1)n+α+1 (1 -λz) n+β (z -λ) n+1 z=xe iϕ dϕ = 1 π ˆπ 0 z (a+1)n+α+1 (1 -λz) n+β (z -λ) n+1 z=xe iϕ
dϕ .

• The standard estimate for contour integrals shows that

1 2πi ˆγ2 f dz ≤ ε max z∈γ 2 z (a+1)n+α+1 (1 -λz) n+β (z -λ) n+1 ≤ ε max z∈γ 2 (1 -λz) β z α+1 z -λ ε (a+1)n 1 -ελ λ -ε n ,
where we have used [START_REF] Garnett | Bounded Analytic Functions[END_REF]Formula (1.11)] for the second inequality. It is clear that the last term goes to 0 as ε → 0.

• We use that z y = e y log(z) = e y|z|+iy arg(z) with the main branch of log and evaluate

ˆγ3 f dz = ˆ0 -x z (a+1)n+α (1 -λz) β z -λ 1 -λz z -λ n z=t+iε dt ε→0 --→ ˆ0 -x |t| (a+1)n+α e ((a+1)n+α)πi |1 -λt| β |t|e πi -λ 1 -λ|t|e πi |t|e πi -λ n dt = e ((a+1)n+α)πi (-1) n+1 ˆ0 -x |t| (a+1)n+α |1 -λt| β |t| + λ 1 + λ|t| |t| + λ n dt = e ((a+1)n+α)πi (-1) n+1 ˆx 0 t (a+1)n+α (1 + λt) β t + λ 1 + λt t + λ n dt.
Similarly we find that

ˆγ4 f dz = - ˆx 0 z (a+1)n+α (1 -λz) β z -λ 1 -λz z -λ n z=-t-iε dt ε→0 --→ - ˆx 0 |t| (a+1)n+α e -((a+1)n+α)πi |1 + λt| β |t|e -πi -λ 1 -λ|t|e -πi |t|e -πi -λ n dt = -e -((a+1)n+α)πi (-1) n+1 ˆx 0 t (a+1)n+α (1 + λt) β t + λ 1 + λt t + λ n dt.
Summing the two terms we find that

1 2πi ˆγ3 ⊕γ 4 f dz ε→0 --→(-1) n+1 sin ((a + 1)n + α)π π ˆx 0 t (a+1)n+α (1 + λt) β t + λ 1 + λt t + λ n dt.
• To evaluate the integral over γ 5 we note that f is holomorphic in a dotted neighborhood around λ. We compute

1 2πi ˆγ5 f dz = s -n 2π ˆπ -π (λ + se iφ ) (a+1)n+α (1 -λ 2 -sλe iϕ ) n+β e -inϕ dϕ.
For s ∈ (0, λ)

(λ + se iφ ) α+(a+1)n = ∞ µ=0 α + (a + 1)n µ λ α+(a+1)n-µ (se iφ ) µ ,
and for s ∈ (0,

1-λ 2 λ ) (1 -λ 2 -sλe iφ ) n+β = ∞ ν=0 n + β ν (1 -λ 2 ) n+β-ν (-λse iφ ) ν ,
and we find that

1 2πi ˆγ5 f dz = s -n 2π ∞ µ=0 ∞ ν=0 α + (a + 1)n µ n + β ν λ α+(a+1)n-µ (-λ) ν s µ (1 -λ 2 ) n+β-ν s ν • ˆπ -π e -i(ν+µ-n)φ dφ 2π .
The last integral is 0 unless ν + µ -n = 0 so that we set ν = n -µ and find

1 2πi ˆγ5 f dz = n µ=0 α + (a + 1)n µ n + β n -µ λ α+(a+1)n-µ (-λ) n-µ (1 -λ 2 ) β+µ = λ α+an (1 -λ 2 ) β n µ=0 α + (a + 1)n µ n + β n -µ (-λ 2 ) n-µ (1 -λ 2 ) µ = λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 ).

Asymptotic analysis of integral representation

3.1. Generalized Fourier integral via the method of stationary phase. This section is devoted to the study of the large n behavior of the integral

ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ when a ∈ [-2λ 1+λ , 2λ 1+λ
]. This is achieved using standard methods from the theory of generalized Fourier integrals. The latter are integrals of the form ˆb a g(t)e inh(t) dt, with g and h are continuous real functions. The integral is perfectly suited for an application of the method of stationary phase. As z a+1 1-λz z-λ = 1 for any z = e iϕ we can write

z a+1 1 -λz z -λ = exp (ih(ϕ))
with a real function h, which for z = e iϕ is defined as

h(ϕ) = -i log z a+1 (1 -λz) z -λ = Arg z a+1 (1 -λz) z -λ .
In this paper log always means the principal branch of the complex logarithm.

Proposition 4. Let λ ∈ (0, 1), a ∈ [-2λ 1+λ , 2λ 1-λ ] and α, β ∈ R. We have the following asymptotic expansion as n → ∞:

(1)

If a ∈ (-2λ 1+λ , 2λ 1-λ ) then 1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = 2 nπ (1 -λ 2 ) β 2 (a + 1) β 2 cos (n + 1)h(ϕ + ) + (α -a)ϕ + + (β -1)ψ+ π 4 [(1 -λ 2 )((a + 2)λ + a)((a + 2)λ -a)] 1 4 + sin ((an + α)π) nπ (1 + λ) β a(1 + λ) + 2λ + O(n -3/2 )
where the parameters ϕ + , ψ ∈ [0, π] are defined via the relations

e iϕ + = a + aλ 2 + 2λ 2 2λ(a + 1) + i 1 - a + aλ 2 + 2λ 2 2λ(a + 1) 2 and 1 -λ 2 a + 1 e iψ = 1 -λz + .
(

) If a = 2λ 1-λ then 1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = (1 -λ) β 3 2/3 Γ(2/3)n 1/3 λ 1/3 (1 + λ) 1/3 1 + O(n -1/3 ) , and if a = -2λ 1+λ then 1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = 1 π cos an + α + 1 6 π Γ(1/3)(1 + λ) β 3 2/3 1 nλ(1 -λ) 1/3 1 + O(n -1/3 ) . 2
Proof. We apply the method of stationary phase [AE] [BO, Chapter 6.5] to the generalized Fourier integral

ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = ˆπ 0 g(ϕ)e inh(ϕ) dϕ,
where for z = e iϕ we define the functions

ϕ → g(ϕ) = z α+1 (1 -λz) β z -λ , ϕ → h(ϕ) = -i log z a+1 (1 -λz) z -λ = Arg z a+1 (1 -λz) z -λ .
Computing derivatives (in the sense of the chain rule) leads to

i dh dz = - 1 z -λ + a + 1 z - λ 1 -λz , i d dz dh dz = 1 (z -λ) 2 - a + 1 z 2 - λ 2 (1 -λz) 2 , i d 3 h dz 3 = - 2 (z -λ) 3 + 2(a + 1) z 3 - 2λ 3 (1 -λz) 3 .
The function h(ϕ) has a stationary point if and only if dh/dz = 0, i.e. iff

a = λ z -λ + λz 1 -λz .
Solving the latter for z gives

z +,-= a + aλ 2 + 2λ 2 2λ(a + 1) ± i 1 - a + aλ 2 + 2λ 2 2λ(a + 1) 2 ∈ ∂D
and we write z +,-= e iϕ +,-with ϕ + ∈ [0, π] and ϕ -∈ (-π, 0]. Only z + is interesting because we integrate over [0, π]. For the second derivative we have that

h (ϕ) = d dϕ dh dz dz dϕ = d 2 h dz 2 dz dϕ 2 + dh dz d 2 z (dϕ) 2 .
We distinguish the two cases 1) a ∈ (-2λ 1+λ , 2λ 1-λ ) and 2) a ∈ -2λ 1+λ , 2λ 1-λ , which are characterized by the presence of a stationary point of order one (h (ϕ + ) = 0 but h (ϕ + ) = 0) in Case 1) and of order two (h

(ϕ + ) = h (ϕ + ) = 0 but h (ϕ + ) = 0) in Case 2).
Case 1) If a ∈ (-2λ 1+λ , 2λ 1-λ ) then the zeros z + = e iϕ + and z -= e iϕ -of h are distinct points located on ∂D with ϕ + ∈ [0, π] and ϕ -∈ (-π, 0]. Plugging in we see that

i d 2 h dz 2 z=z + = (1 -λ 2 )(1 -z 2 + )λ z + (z + -λ) 2 (1 -λz + ) 2 so that h (ϕ + ) > 0.
To find the asymptotics we apply a standard result by A. Erdélyi [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] (see also F. Olver [START_REF] Olver | Error bounds for stationary phase approximations[END_REF]Theorem 1] for a more explicit form), which however requires that the stationary point is an endpoint of the interval of integration. Hence we begin by splitting ˆπ 0 g(ϕ)e inh(ϕ) dϕ = ˆϕ+ 0 g(ϕ)e inh(ϕ) dϕ + ˆπ ϕ + g(ϕ)e inh(ϕ) dϕ.

For the second integral [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] (see also [START_REF] Olver | Error bounds for stationary phase approximations[END_REF] Theorem 1]) gives

ˆπ ϕ + g(ϕ)e inh(ϕ) dϕ = 1 2 Γ(1/2)k(0)e i π 4 n -1/2 e inh(ϕ + ) + 1 2 Γ(1)k (0)e i π 2 n -1 e inh(ϕ + ) - i n e inh(π) g(π) h (π) + O n 3/2 , with k(0) = 2 1/2 g(ϕ + ) (h (ϕ + )) -1/2 and k (0) = 2 h (ϕ + ) g (ϕ + ) - 2 h (ϕ + ) h (3) (ϕ + ) 3h (ϕ + ) g(ϕ + ).
For the first integral ´ϕ+ 0 g(ϕ)e inh(ϕ) dϕ we change the variable of integration ϕ → -ϕ as suggested in [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]page 23]. We get

ˆϕ+ 0 g(ϕ)e inh(ϕ) dϕ = ˆ0 -ϕ + g(-ϕ)e inh(-ϕ) dϕ.
Applying [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] (see also [START_REF] Olver | Error bounds for stationary phase approximations[END_REF] Theorem 1]) gives

ˆ0 -ϕ + g(-ϕ)e inh(-ϕ) dϕ = 1 2 Γ(1/2)k(0)e i π 4 n -1/2 e inh(ϕ + ) + 1 2 Γ(1)k (0)e i π 2 n -1 e inh(ϕ + ) - i n e inh(0) g(0) h (0) + O n 3/2 with k(0) = 2 1/2 g(ϕ + ) (h (ϕ + )) -1/2 and k (0) = - 2 h (ϕ + ) g (ϕ + ) + 2 h (ϕ + ) h (3) (ϕ + ) 3h (ϕ + ) g(ϕ + ).
Observing that h(0) = 0 while h(π) = aπ, g(0) = (1-λ) β-1 while g(π) = e i(α+1)π (1+λ) β-1 , and h (0

) = a(1-λ)-2λ 1-λ while h (π) = -a(1+λ)+2λ 1+λ we get - i n e inh(π) g(π) h (π) = i n e i(an+α)π (1 + λ) β a(1 + λ) + 2λ while - i n e inh(0) g(0) h (0) = - i n (1 -λ) β a(1 -λ) -2λ
and we conclude that ˆπ 0 g(ϕ)e inh(ϕ) dϕ

= Γ(1/2) 2 1/2 g(ϕ + ) (h (ϕ + )) -1/2 e i π 4 n -1/2 e inh(ϕ + ) + i n e i(an+α)π (1 + λ) β a(1 + λ) + 2λ - i n (1 -λ) β a(1 -λ) -2λ + O n -3/2 = e inh(ϕ + )+i π 4 z α+1 + (1 -λz + ) β z + -λ 2|z + -λ| 4 nλ(1 -λ 2 )|1 -z 2 + | 1/2 Γ(1/2) + 1 n e i((an+α)π+ π 2 ) (1 + λ) β a(1 + λ) + 2λ - i n (1 -λ) β a(1 -λ) -2λ + O n -3/2 .
We set ψ = arg (1 -λz + ) and get

e inh(ϕ + )+i π 4 z α+1 + (1 -λz + ) β z + -λ 2|z + -λ| 4 nλ(1 -λ 2 )|1 -z 2 + | 1/2 Γ(1/2) = √ 2π|1 -λz + | β-1+2 nλ(1 -λ 2 )|1 -z 2 + | e i(nh(ϕ + )+ π 4 +(α-a)ϕ + +(β-1)ψ+h(ϕ + )) 1 + O(n -1 ) = 2π n (1 -λ 2 ) β 2 -1 4 (a + 1) β 2 e i((n+1)h(ϕ + )+(α-a)ϕ + +(β-1)ψ+ π 4 ) (((a + 2)λ + a)((a + 2)λ -a)) 1 4 1 + O(n -1 ) ,
where we made use of

|1 -λz + | = |z + -λ| = 1 -λ 2 a + 1 and |z 2 + -1| = 2 1 - a + aλ 2 + 2λ 2 2λ(a + 1) 2 = (1 -λ 2 )((a + 2)λ + a)((a + 2)λ -a) λ(a + 1) .
Taking the real part of ´π 0 g(ϕ)e inh(ϕ) dϕ completes the proof of Theorem 4, point 1) since the real part of 1 n e i((an+α)π+ π 2 ) (1+λ) β a(1+λ)+2λ is equal to sin ((an + α)π) n [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] at the second order in this case yields the asymptotic behavior ˆπ 0 g(ϕ)e inh(ϕ) dϕ = g(ϕ + )e inh(ϕ

(1 + λ) β a(1 + λ) + 2λ . Case 2) If a ∈ {-2λ 1+λ , 2λ 1-λ } then h has a unique zero. If a = 2λ 1-λ then z + = 1 and h(0) = h (0) = h (0) = 0, while h (3) (0) = 2λ(1 + λ) (1 -λ) 3 . Applying [
+ )+i π 6 6 n|h (3) (ϕ + )| 1/3 Γ(1/3) 3 1 + O(n -1/3 ) .
Direct computation shows that ˆπ 0 g(ϕ)e inh(ϕ) dϕ = z α+1

+ (1 -λz + ) β z + -λ e i π 6 z a+1 + (1 -λz + ) z + -λ n 3(1 -λ) 3 nλ(1 + λ) 1/3 Γ(1/3) 3 1 + O(n -1/3 ) = e i π 6 (1 -λ) β 3 nλ(1 + λ) 1/3 Γ(1/3) 3 1 + O(n -1/3 ) ,
and we conclude that ϕ) dϕ so that the saddle point ϕ + = -π is the left endpoint in the interval of integration; indeed we have

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = (1 -λ) β 3 2/3 Γ(2/3) 1 nλ(1 + λ) 1/3 1 + O(n -1/3 ) . If a = -2λ 1+λ then z + = -1 Observe first that ˆ0 -π g(ϕ)e inh(ϕ) dϕ = ˆπ 0 g(ϕ)e inh(
h (-π) = h (-π) = 0, h (3) (-π) = - 2λ(1 -λ) (1 + λ) 3 .
Applying again [START_REF] Erdélyi | Asymptotic representations of Fourier integrals and the method of stationary phase[END_REF]Theorem 4] in this case yields the asymptotic behavior ˆ0 -π g(ϕ)e inh(ϕ) dϕ = g(-π)e inh(-π)-i π

6 6 n|h (3) (-π)| 1/3 Γ(1/3) 3 1 + O(n -1/3 ) . Now g(-π) = e -iπ(α+1) (1 + λ) β -(1 + λ) = (1 + λ) β-1 e -iαπ h(-π) = Arg e -iπ(a+1) (1 + λ) -(1 + λ) = -iaπ.
This gives

ˆ0 -π g(ϕ)e inh(ϕ) dϕ = (1 + λ) β-1 e -iαπ e -inaπ-i π 6 3(1 + λ) 3 nλ(1 -λ) 1/3 Γ(1/3) 3 1 + O(n -1/3 )
and we conclude that

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=e iϕ dϕ = 1 π cos an + α + 1 6 π Γ(1/3)(1 + λ) β 3 2/3 1 nλ(1 -λ) 1/3 1 + O(n -1/3 ) .
3.2. The Laplace type integral via Laplace's method. This section is devoted to the study of the large n behavior of the integral

ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt,
which contributes to the asymptotics of Jacobi polynomials with varying parameters iff α + an is not an integer. This is achieved using standard methods [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF][START_REF] Olver | Asymptotics and Special Functions[END_REF] from the theory of Laplace-type integrals. The latter are integrals of the form

ˆ1 0 f (t)e nh(t) dt,
where f and h are real continuous functions. In essence the idea is that if the real continuous function h has a maximum at a point ξ ∈ [0, 1] and if f (ξ) = 0 then as n grows large only values in an immediate neighborhood of ξ should contribute to the integral, see [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF][START_REF] Olver | Asymptotics and Special Functions[END_REF]. To apply the method we introduce for α, β ∈ R, a > -1 and λ ∈ (0, 1) (overwriting previous notation) the real functions

t → f (t) = (1 + λt) β t α t + λ , t ∈ [0, 1] t → g(t) = t a+1 1 + λt λ + t , t ∈ [0, 1] t → h(t) = ln (g(t)), t ∈ (0, 1].
With this notation we can write out the integral in Laplace form

ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt = ˆ1 0 f (t) (g(t)) n ds = ˆ1 0 f (t)e nh(t) dt.
The first step is to gain a picture about the behavior of h on (0, 1]. Computing derivatives we find

h (t) = a + 1 t - 1 t + λ + λ 1 + λt , h (t) = - a + 1 t 2 + 1 (t + λ) 2 - λ 2 (1 + λt) 2 , h (t) = 2a + 2 t 3 - 2 (t + λ) 3 + 2λ 3 (1 + λt) 3 .
We note that h (t) = 0 holds iff

a = t λ + t - λt 1 + λt -1.
For t this yields two solutions

t +,-= - a + aλ 2 + 2λ 2 2λ(a + 1) ± a + aλ 2 + 2λ 2 2λ(a + 1) 2 -1,
which are real if and only if (a ≥ 2λ 1-λ or a ≤ -2λ 1+λ ).

Lemma 5. Let h(t) = ln t a+1 1+tλ t+λ , t ∈ (0, 1], λ ∈ (0, 1) and let a > -1. For h we have that lim t→0 + h(t) = -∞ and h(1) = 0 and the following properties

(1) If a > -2λ
1+λ then h(t) ≤ 0 and h (t) > 0 for t ∈ (0, 1].

(2) If a = -2λ 1+λ then t +,-= 1 and h(t) < 0 and h (t) > 0 for t ∈ (0, 1). Further h (1) = h (1) = 0 and h

(3) (1) = 2λ(1-λ) (1+λ) 3 . (3) If a ∈ (-1, -2λ
1+λ ) then h has a unique maximum at t -∈ (0, 1) with h(t -) > 0,

h (t -) = 0, h (t -) = λ t - - 1 (λ + t -) 2 + 1 (1 + λt -) 2 < 0.
For illustration Figure 3.1 depicts the three cases of the lemma. Proof. The proof follows from the formulas for the derivatives of h.

(1) We have already seen that h (t) = 0 holds iff

a = t λ + t - λt 1 + λt -1.
The function t λ+t -λt 1+λt -1 is increasing on [0, 1] and achieves the maximum -2λ 1+λ at t = 1. Hence, if a > -2λ 1+λ then the above equation has no solution t * ∈ (0, 1]. In other words h has no zeros on the interval (0, 1] i.e. h is either positive or negative on the whole interval. Since h(0) = -∞ and h(1) = 0 it follows that h is strictly increasing on [0, 1) and subsequently we also have that h(t) ≤ 0 on (0, 1].

(2) Plugging a = -2λ 1+λ into above formulas shows that t +,-= 1 and that h (1) = h (1) = 0 and h (1) = 2λ(1-λ) (1+λ) 3 . As in (1) we can conclude that t = 1 is the only zero of h in (0, 1] so that h is strictly positive. Furthermore it holds that h(t) ≤ 0 on (0, 1].

(3) If a ∈ (-1, -2λ 1+λ ) then t -∈ (0, 1) and t + > 1 so that h has a unique zero t -∈ (0, 1) with

a = t - λ + t - - λt - 1 + λt - -1.
Plugging this into our formula for h (t) we find that

h (t -) = λ t - - 1 (λ + t -) 2 + 1 (1 + λt -) 2 < 0.
We conclude that t -is a local maximum of h and that h is decreasing in a neighborhood of t -. Since lim t→0+ h(t) = -∞ and h(1) = 0 and there are no other critical points of h we conclude that h(t -) is a global maximum of h and therefore it is positive. Proposition 6. Let λ ∈ (0, 1), a > -1 and let α, β > -1. We have the following asymptotic behavior as n → ∞:

(1)

If a > -2λ 1+λ then ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt = 1 n (1 + λ) β λ(a + 2) + a 1 + O(n -1 ) .
(

) If a = -2λ 1+λ then ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt = 1 n 1/3 Γ(1/3) 3 2/3 (1 + λ) β (λ(1 -λ)) 1/3 1 + O(n -1/3 ) . (3) If a ∈ (-1, -2λ 1+λ ) then g(t -) > 1 and ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ n dt ∼ (1 + λt -) β+1 t α -(g(t -)) n 2πt - nλ ((1 + λt -) 2 -(λ + t -) 2 ) . 2
Proof. For properties of h we refer to the respective points of Lemma 5. We employ standard Laplace methods to study the asymptotic behavior

ˆ1 0 (1 + λt) β (-t) α t + λ t (a+1) 1 + λt t + λ n dt = ˆ0 -1 (1 -λt) β t α λ -t (-t) (a+1) 1 -λt t -λ n dt = ˆ0 -1 f (t)e -n h(t) dt,
where

f (t) = f (-t) = (1 -λt) β (-t) α λ -t , h(t) = -h(-t) = -ln (-t) a+1 1 -λt λ -t .
(1) If a > -2λ 1+λ then h(-1) = 0, lim t→0 + h(t) = +∞ and for t ∈ [-1, 0) that h(t) ≥ 0 and h (t) > 0. Applying a result by A. Erdélyi (1956) [START_REF] Olver | Asymptotics and Special Functions[END_REF]Theorem 7.1 page 81] we get

ˆ0 -1 f (t)e -n h(t) dt = f (-1) n h (-1) e n h(-1) 1 + O(n -1 ) = 1 n (1 + λ) β λ(a + 2) + a 1 + O(n -1 ) which completes the proof. (2) If a = -2λ
1+λ then h is positive and strictly increasing on

[-1, 0) with h(-1) = 0, lim t→0 -h(t) = ∞. Moreover h (-1) = h (-1) = 0 while h(3) (-1) = h (3) (1) = 2λ(1-λ) (1+λ) 3 > 0. Applying again A. Erdélyi's result [FO0, Theorem 7.1 page 81] we get ˆ0 -1 f (t)e -n h(t) dt = f (-1)e -n h(-1) Γ(1/3) 3 3! n h(3) (-1) 1/3 1 + O(n -1/3 ) = (1 + λ) β-1 n 1/3 Γ(1/3) 3 2/3 1 + λ (λ(1 -λ)) 1/3 1 + O(n -1/3 ) = 1 n 1/3 Γ(1/3) 3 2/3 (1 + λ) β (λ(1 -λ)) 1/3 1 + O(n -1/3 ) .
(3) If a ∈ (-1, -2λ 1+λ ) then t -is the unique maximum of h on (0, 1) with g(t -) = e h(t -) > 1. Applying standard results about Laplace integrals [START_REF] Bender | Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory[END_REF]formula (6.4.19c) page 267] we obtain ˆ1 0

(1 + λt)

β t α t + λ t (a+1) 1 + λt t + λ n dt ∼ (1 + λt -) β t α - t -+ λ 2π -nh (t -) (g(t -)) n , as n → +∞. Moreover 2π -nh (t -) = (λ + t -)(1 + λt -) 2πt - nλ ((1 + λt -) 2 -(λ + t -) 2 )
and the result follows.

Simple exponential upper estimates

In Lemma 3 we chose to not specify the parameter x ∈ (λ, 1/λ). This allows us to tune the parameter to find simple exponential bounds for Jacobi polynomials for a ∈ (-1, -2λ 1+λ ) ∪ ( 2λ 1-λ , ∞). Proposition 7. Let α, β > -1, a > -1, λ ∈ (0, 1) and n be a positive integer.

(1) We assume that α + an is an integer.

(a) If a > 2λ 1-λ then there is x * ∈ (λ, 1) such that

|λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 )| ≤ (x * ) α-a ψ β (λx * ) (x * ) a+1 1 -λx * x * -λ n+1 and (x * ) a+1 1-λx * x * -λ < 1 where ψ β (u) = (1 + u) β-1 if β ≥ 1 (1 -u) β-1 , if β < 1 .
x * is computed explicitly below in (4.2). (b) If a < -2λ 1+λ then for any x ∈ (1, 1/λ)

|λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 )| ≤ x α-a ψ β (λx) x a+1 1 -λx x -λ n+1 and x a+1 1-λx x-λ < 1.
(2) We assume that α + an is not an integer.

(a) If a > 2λ 1-λ then there is x * ∈ (0, 1) such that |λ an+α (1 -λ 2 ) β P (an+α, β) n

(1 -2λ 2 )| ≤ (x * ) α-a ψ β (λx * ) (x * ) a+1 1 -λx * x * -λ n+1 + ˆ1 0 (1 + λt) β t α t + λ dt (x * ) a+1 1 + λx * x * + λ n . and (x * ) a+1 1-λx * x * -λ < 1, (x * ) a+1 1+λx * x * +λ < 1. (b) If a < -2λ 1+λ λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 ) ∼ (1 + λt -) β+1 t α -(g(t -)) n 2πt - nλ ((1 + λt -) 2 -(λ + t -) 2 )
where t -∈ (0, 1) is computed explicitly below in (3.2) and g(t

-) = t a+1 - 1+λt - λ+t -> 1.
To start with we find upper estimates for

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=xe iϕ dϕ .
Lemma 8. We have the following exponential upper bounds.

(1) If a > 2λ 1-λ then there is x * ∈ (λ, 1) such that (x * ) a+1 1-λx * x * -λ ∈ (0, 1) and

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=x * e iϕ dϕ ≤ (x * ) α-a ψ β (λx * ) (x * ) a+1 1 -λx * x * -λ n+1 where ψ β (u) = (1 + u) β-1 if β ≥ 1 (1 -u) β-1 , if β < 1 . x * is computed explicitly below in (4.2).
(2) If -1 < a < -2λ 1+λ then for any x ∈ (1, 1/λ) we have x a+1 1-λx x-λ ∈ (0, 1) and

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=xe iϕ dϕ ≤ x α-a ψ β (λx) x a+1 1 -λx x -λ n+1 .
Proof. Clearly

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=xe iϕ dϕ ≤ max |z|=x z α+1 (1 -λz) β z -λ max |z|=x |z a+1 1 -λz z -λ | n .
For z = xe iϕ we have

max |z|=x z a+1 1 -λz z -λ = x a+1 max -π≤ϕ≤π 1 -λxe iϕ xe iϕ -λ .
Direct computation shows that

∂ ∂ϕ | 1 -λxe iϕ xe iϕ -λ | 2 = - 2(1 -x 2 )(1 -λ 2 )λx sin ϕ |xe iϕ -λ| 2 ,
so the corresponding max is attained either at ϕ = 0 or at ϕ = ±π. Thus max

-π≤ϕ≤π | 1-λxe iϕ xe iϕ -λ | is either equal to | 1-λx x-λ | or to 1+λx x+λ . Since x ∈ (λ, 1/λ) we have | 1-λx x-λ | = 1-λx x-λ and 1 -λx x -λ - 1 + λx x + λ = 2λ(1 -x 2 ) x 2 -λ 2 > 0. This gives max -π≤ϕ≤π | 1-λxe iϕ xe iϕ -λ | = 1-λx x-λ and thus max |z|=x |z a+1 1 -λz z -λ | = x a+1 1 -λx x -λ , but also max |z|=x |z α+1 (1 -λz) β z -λ | = x α+1 max |z|=x | (1 -λz) β z -λ | ≤ x α+1 ψ β (λx) max |z|=x | 1 -λz z -λ | = x α+1 ψ β (λx) 1 -λx x -λ ,
and as a consequence

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=xe iϕ dϕ ≤ x α+1 ψ β (λx) 1 -λx x -λ x a+1 1 -λx x -λ n ≤ x α-a ψ β (λx) x a+1 1 -λx x -λ n+1 .
We put g(x) = x a+1 1-λx x-λ . The derivative of g with respect to x is zero if and only if the variables λ, a, x satisfy

a = λ x -λ + λx 1 -λx .
Solving the latter for x gives:

x +,-= a + aλ 2 + 2λ 2 2λ(a + 1) ± a + aλ 2 + 2λ 2 2λ(a + 1) 2 -1,
which is real if and only if (a ≥ 2λ 1-λ or a ≤ -2λ 1+λ ). Moreover direct computation shows that (4.1)

g (x +,-) = x a+1 +,- λ(1 -x 2 +,-)(1 -λ 2 ) x +,-(x +,--λ) 3 (1 -λx +,-)
and the fact that x +,-is a max or min will depend on its position with respect to 1, λ and 1/λ.

1) We first prove that if a > 2λ 1-λ then x + ∈ (1, 1/λ) while x -∈ (λ, 1) and

(4.2) x * = x -= a + aλ 2 + 2λ 2 2λ(a + 1) - a + aλ 2 + 2λ 2 2λ(a + 1) 2 -1 satisfies g(x * ) < 1. We note that x +,-(a = 2λ 1-λ ) = 1, x +,-(a = -2λ 1+λ ) = -1 while x + (a → ∞) → 1 λ and x -(a → ∞) → λ. The function a → y(a) = a+aλ 2 +2λ 2
2λ(a+1) is monotonically increasing on R for λ ∈ (0, 1) and the function f + (y) = y + y 2 -1 is increasing for y > 1. This shows that x + (a) is increasing for a > 2λ 1-λ so that x + ∈ (1, 1/λ). In particular x → g (x) vanishes on (λ, 1] if and only if x = x -. Moreover the fact that x -∈ (λ, 1) taken together with formula (4.1) gives g (x -) < 0 and x -is a min for g on (λ, 1). Taking into account lim x→λ + g(x) = +∞ and g(1) = 1, g is strictly decreasing on (λ, x -], strictly increasing on [x -, 1] and admits a unique minimum x * = x -on (λ, 1] which satisfies g(x * ) < 1.

2) We prove that if a < -2λ 1-λ then x +,-< 0 and for any x ∈ (1, 1/λ) we have g(x) < 1.

Indeed we observe first that

-1 < a < - 2λ 1 -λ < - 2λ 2 1 + λ 2 because 2λ 1+λ -2λ 2 1+λ 2 = 2λ(1-λ) (1+λ)(1+λ 2 ) > 0.
In particular a + aλ 2 + 2λ 2 2λ(a + 1) < 0 and thus x -< 0. Moreover

a + aλ 2 + 2λ 2 2λ(a + 1) 2 -1 < | a + aλ 2 + 2λ 2 2λ(a + 1) | so that a + aλ 2 + 2λ 2 2λ(a + 1) + a + aλ 2 + 2λ 2 2λ(a + 1) 2 -1 < 0.
In particular g does not vanish on (λ, 1/λ] with lim x→λ + g(x) = +∞, g(1) = 1 and g(1/λ) = 0: as a consequence g is strictly decreasing on (1, 1/λ) and satisfies 0 < g(x) < 1 for any x ∈ (1, 1/λ).

We conclude by proving Proposition 7.

Proof of Proposition 7. Since (1) is clear from Lemma 8 we prove (2).

(2) If a > 2λ 1-λ we apply Lemma 8 point (1) and choose x = x * ∈ (λ, 1) such that (x * ) a+1 1-λx *

x * -λ ∈ (0, 1). We have and g(t -) > 1.

1 π ˆπ 0 z α+1 (1 -λz) β z -λ z a+1 1 -λz z -λ n z=x * e iϕ dϕ ≤ (x * ) α-a ψ β (λx * ) (x * ) a+1 1 -λx * x * -λ n+1 where ψ β (u) = (1 + u) β-1 if β ≥ 1 (1 -u)

Proof of Theorem 2

We add this section for completeness. We collect our results on the two integrals and write up a proof of the main theorem. Recall that according to Lemma 3 we have λ an+α (1 -λ 2 ) β P (an+α, β) n

(1 -2λ 2 ) = 1 π ˆπ 0 z α+1 (1 -λz) β z -λ

Figure 3 . 1 .

 31 Figure3.1. This figure depicts the situation described in Lemma 5. In all plots λ = 0.3 is chosen. A) corresponds to point (1) in the lemma and graph shows, a = 0.9 > -6/13. B) corresponds to point (2) in the lemma and graphs show a = -6/13. C) corresponds to point (3) in the lemma and graphs show a = -0.9 < -6/13.

  Similarly x -∈ (λ, 1) and x -(a) is decreasing for a > 2λ 1-λ since the function f -(y) = y -y 2 -1 is decreasing for y > 1. For illustration Figure 4.1 depicts graphs of f + (y) = y + y 2 -1 and f -(y) = y -y 2 -1.

Figure 4 . 1 .

 41 Figure 4.1. Plot of functions f + (y) = y + y 2 -1 (red) and f -(y) = y -y 2 -1 (blue).

  We choose any x such that x ∈ (1, min(1/λ, t + )) where and observe that the second critical point t + for g (the first one being t -∈ (λ, 1) see Lemma 5 point (3)) satisfies t + > 1. Indeed nλ ((1 + λt -) 2 -(λ + t -) 2 ) ,

		t + = -	a + aλ 2 + 2λ 2 2λ(a + 1)	+	a + aλ 2 + 2λ 2 2λ(a + 1)
	t + -1 =	a + aλ 2 + 2λ 2 2λ(a + 1)		2	-1 -	(1 + λ)(a(1 + λ) + 2λ) 2λ(a + 1)	> 0
	since a(1 + λ) + 2λ < 0. Then we write
	ˆx 0 = ˆ1 0 (1 + λt) β t α t + λ (1 + λt) β t α t (a+1) 1 + λt t + λ t + λ t (a+1) 1 + λt n t + λ	dt n	dt +	ˆx 1	(1 + λt) β t α t + λ	t (a+1) 1 + λt t + λ
	t+λ is strictly increasing on [0, 1] so t → t (a+1) 1+λt ∼ ˆ1 0 (1 + λt) β t α t + λ t (a+1) 1 + λt t + λ
	where		ˆx * 0	(1 + λt) β t α t + λ			t (a+1) 1 + λt t + λ
		t -= -	a + aλ 2 + 2λ 2 2λ(a + 1)	-	a + aλ 2 + 2λ 2 2λ(a + 1)
	n+1 (x If a < -2λ + ˆ1 0 (1 + λt) β t α t + λ dt 1+λ we apply Lemma 8 point (2). For any x ∈ (1, 1/λ) we have x a+1 1-λx x-λ ∈ (0, 1)
	and
	|	1 π	ˆπ 0	z α+1 (1 -λz) β z -λ	z a+1 1 -λz z -λ	n	z=xe iϕ	dϕ |

β-1 , if β < 1 . Moreover according to Lemma 5 point (1) g :

n dt ≤ (x * ) a+1 1 + λx * x * + λ n ˆx * 0 (1 + λt) β t α t + λ dt with (x * ) a+1 1+λx * x * +λ < 1. Thus |λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 )| ≤ (x * ) α-a ψ β (λx * ) (x * ) a+1 1 -λx * x * -λ * ) a+1 1 + λx * x * + λ n . ≤ x α-a ψ β (λx) x a+1 1 -λx x -λ n+1 . 2 -1 n dt

and the asymptotic (exponential) behavior of the first integral is given in Lemma 5 point (2) while the second one is O(n -1 ) because x has been chosen so that g does not vanish on

[1, x]. Thus λ an+α (1 -λ 2 ) β P (an+α, β) n (1 -2λ 2 ) n dt ∼ (1 + λt -) β+1 t α -(g(t -)) n 2πt - 2 -1

The work is supported by Russian Science Foundation grant 14-41-00010.

Proof of Theorem 2. (1) If a ∈ (-2λ 1+λ , 2λ 1-λ ) we choose x = 1 and conclude by combining Proposition 6 point (1) and Proposition 4 point (1).

(2) If a = 2λ 1-λ we choose again x = 1 and conclude by combining Proposition 6 point (2) andProposition 4 point (2). If a = -2λ 1+λ we choose x = 1. According to Proposition 6 point (2) we have

where C(λ, β) = Γ(1/3)

The result follows by subtracting the two above equalities. ( 3) and ( 4) are direct consequences of Proposition 7.