Rémi Carles

Dmitry Pelinovsky

We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in H 1 (R) with conserved L 2 norm and energy, we construct a weak global solution of the log-KdV equation in a subset of H 1 (R). This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided uniqueness and continuous dependence of the constructed solution holds.

Furthermore, we study the linearized log-KdV equation at the Gaussian solitary wave and prove that the associated linearized operator has a purely discrete spectrum consisting of simple purely imaginary eigenvalues in addition to the double zero eigenvalue. The eigenfunctions, however, do not decay like Gaussian functions but have algebraic decay. Using numerical approximations, we show that the Gaussian initial data do not spread out but produce visible radiation at the left slope of the Gaussian-like pulse in the time evolution of the linearized log-KdV equation.

On the orbital stability of Gaussian solitary waves in the log-KdV equation

Solitary waves in anharmonic chains with Hertzian interaction forces are modelled by the Fermi-Pasta-Ulam (FPU) lattices with non-smooth nonlinear potentials [START_REF] Nesterenko | Dynamics of heterogeneous materials[END_REF]. Recently, the FPU lattice equations in the limit of small anharmonicity of the Hertzian interaction forces were reduced to the following logarithmic Korteweg-de Vries (log-KdV) equation [START_REF] Chatterjee | Asymptotic solution for solitary waves in a chain of elastic spheres[END_REF][START_REF] James | Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials[END_REF]:

v t + v xxx + (v log |v|) x = 0, (x, t) ∈ R × R. (1.1)
Here and in what follows, the subscripts denote the partial derivatives. The log-KdV equation (1.1) has a two-parameter family of Gaussian solitary waves

v(x, t) = e c v G (x -ct -a), a, c ∈ R, (1.2)
where v G is a symmetric standing wave given by v G (x) := e The ultimate goal of this work is to prove the nonlinear orbital stability of Gaussian solitary waves (1.2) in the log-KdV equation (1.1). The main problem is, of course, the limited smoothness of the log-KdV equation, where the nonlinearity f (v) = v log |v| is continuous but not differentiable at v = 0, whereas the energy

E(v) = 1 2 R (v x ) 2 -v 2 log |v| dx + 1 4 R v 2 dx (1.4)
is not a C 2 functional at v = 0. Although E(v) is not C 2 at v = 0, the second variation of E(v) at v G is well determined by the Schrödinger operator with a harmonic potential L := -∂ 2

x + 1 4 (x 2 -6).

(1.5)

Note that the spectrum of L in L 2 (R) consists of equally spaced simple eigenvalues σ(L) = {-1, 0, 1, 2, . . .}, which include exactly one negative eigenvalue with the eigenvector v G (defined without normalization). Therefore, E(v) is not convex at v G in a subspace of H 1 (R). Nevertheless, the second variation of E(v) at v G given by E c (u) = 1 2 Lu, u L 2 is positive in the constrained space

X c := u ∈ H 1 (R) ∩ L 2 1 (R) : v G , u L 2 = 0 , (1.6)
where L 2 1 (R) := {u ∈ L 2 (R) : xu ∈ L 2 (R)}. At the linearized approximation, this constraint fixes v 2 L 2 at v G 2 L 2 . Based on these facts, the linear orbital stability of the Gaussian solitary wave v G can be deduced in the following sense.

Consider the time evolution of the linearized log-KdV equation at the Gaussian wave v G , u t = ∂ x Lu, t > 0, u| t=0 = u 0 .

(1.7)

Note that the quadratic energy function E c (u) is constant in time for smooth solutions of the linearized log-KdV equation (1.7). We say that the Gaussian solitary wave v G is linearly orbitally stable in H 1 (R), if for every u 0 ∈ X c , there exists a unique global solution u of the linearized log-KdV equation (1.7) in a subspace of L ∞ (R, H 1 (R)) which satisfies the following bound

u(•, t) H 1 C( u 0 H 1 , u 0 L 2 1 ), t ∈ R, (1.8)
for some t-independent positive constant C that depends on the initial norms u 0 H 1 and u 0 L 2 1 . The following theorem is based on the fact that the conserved quantity E c (u) is positive if u ∈ X c and controls the squared H 1 norm of the solution u in time t. Although this theorem was not formulated in [START_REF] James | Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials[END_REF], it can be deduced from the arguments developed in this work, which rely on symplectic projections and the energy method. Due to the scaling invariance (1.2), the same stability result holds for all values of parameter a and c. Theorem 1.1. [START_REF] James | Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials[END_REF] Gaussian solitary wave v G of the log-KdV equation (1.1) is linearly orbitally stable in H 1 (R).

To extend this result to the proof of nonlinear orbital stability of Gaussian solitary waves, we need local and global well-posedness of the log-KdV equation (1.1). In a similar context of the log-NLS equation, global well-posedness and orbital stability of the Gaussian solitary waves were proved by Cazenave and Haraux [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] and Cazenave [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] (these results were later summarized in the monograph [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Section 9.3]).

The idea from [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] is to approximate the logarithmic nonlinearity by a smooth nonlinearity, to construct a sequence of global bounded solutions of the regularized system in C(R, H 1 (R)), and then to prove convergence of a subsequence to the limit, which solves the original log-NLS equation in a weak sense. Uniqueness of the weak solution does not come in this formalism for free, but it can be proved with an additional trick involving log-nonlinearity [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Lemma 9.3.5], which is based on the bound (2.23) below. Similar ideas were recently used by Carles and Gallo [START_REF] Carles | Finite time extinction by nonlinear damping for the Schrödinger equation[END_REF] for analysis of the NLS equation with power-like nonlinearity and a sublinear damping term (see also [START_REF] Carles | Finite time extinction for nonlinear Schrödinger equation in 1D and 2D[END_REF] for various extensions).

While trying to adopt the programme above to the log-KdV equation (1.1), we come to two main difficulties. The first one is that local solutions of the generalized KdV equation with smooth nonlinearities exist in the space C([-t 0 , t 0 ], H s (R)) for s > 3 2 (see [START_REF] Kato | On the Korteweg-de Vries equation[END_REF][START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF] for two independent proofs of these results). Therefore, to push s to lower values, in particular, to s = 1, we need to adopt the formalism of Kenig, Ponce, and Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF], which was originally developed to the KdV equation with integer powers. The second difficulty is that the proof of uniqueness requires us to consider more restrictive solutions of the log-KdV equation than the ones constructed in the proof of existence. As a result, the solutions we establish in a subspace of H 1 (R) have nondecreasing L 2 norm and energy. Unlike the case of weak solutions of the NLS equation (e.g., treated in [START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF]), we cannot establish even the L 2 conservation in the log-KdV equation (1.1). See Remark 2.4 below for further details.

To make sense of the energy (1.4) of the log-KdV equation (1.1), we shall work with functions in the class

X := v ∈ H 1 (R) : v 2 log |v| ∈ L 1 (R) .
(1.9)

The following theorem gives the main result on the existence of weak solutions of the log-KdV equation (1.1) in the energy space X.

Theorem 1.2. For any v 0 ∈ X, there exists a global solution v ∈ L ∞ (R, X) of the log-KdV equation (1.1) such that

v(t) L 2 v 0 L 2 , E(v(t)) E(v 0 ), for all t ∈ R. (1.10) Moreover, if ∂ x log |v| ∈ L ∞ ((-t 0 , t 0 ) × R), then the solution v exists in C((-t 0 , t 0 ), X)
, is unique for every t ∈ (-t 0 , t 0 ), depends continuously on the initial data v 0 ∈ X, and satisfies conservation of v(t) L 2 and E(v(t)) for all t ∈ (-t 0 , t 0 ).

Unfortunately, ∂ x log |v| is unbounded as |x| → ∞ for the Gaussian solitary wave v G . Therefore, it is unclear from Theorem 1.2 if uniqueness and continuous dependence hold for such Gaussian solutions. Convexity of the energy functional E(v) at v G and global well-posedness of the Cauchy problem for the log-KdV equation (1.1) in X are the two main ingredients of the nonlinear orbital stability of the Gaussian solitary wave v G in H 1 (R). Because of limitations in Theorem 1.2, we can only obtain the conditional nonlinear orbital stability, where the condition is that the global solution v ∈ L ∞ (R, X) of the log-KdV equation (1.1) constructed in Theorem 1.2 is unique and depends continuously on the initial data v 0 ∈ X.

To define the nonlinear orbital stability, we use the standard theory (see [START_REF] Pava | Nonlinear dispersive equations[END_REF] for review of this theory). We say that the Gaussian solitary wave v G is orbitally stable in H 1 (R) if for any ε > 0 there exists δ > 0 such that for any

v 0 ∈ X ⊂ H 1 (R) satisfying v 0 -v G H 1 δ, (1.11)
there exists a unique solution v ∈ C(R, X) of the log-KdV equation (1.1) satisfying

inf a∈R v(•, t) -v G (• + a) H 1 ε, (1.12)
for all t ∈ R. The following conditional orbital stability result follows from Theorems 1.1 and 1.2 with the standard arguments [START_REF] Pava | Nonlinear dispersive equations[END_REF].

Corollary 1.3. Gaussian solitary wave v G of the log-KdV equation (1.1) is orbitally stable in H 1 (R) under condition that the solution v ∈ L ∞ (R, X) in Theorem 1.2 is unique and depends continuously on the initial data v 0 ∈ X.

Since Corollary 1.3 does not give a proper nonlinear orbital stability result, we shall also look at the orbital stability problem from a different point of view. We first inspect properties of the linearized operator ∂ x L in the linearized log-KdV equation (1.7). The following theorem gives the spectral stability of the Gaussian solitary wave v G with precise characterization of eigenvalues and eigenvectors of the linear operator ∂ x L in L 2 (R).

Theorem 1.4. The spectrum of ∂ x L in L 2 (R) is purely discrete and consists of a double zero eigenvalue and a symmetric sequence of simple purely imaginary eigenvalues {±iω n } n∈N such that 0 < ω 1 < ω 2 < ... and ω n → ∞ as n → ∞. The double zero eigenvalue corresponds to the Jordan block

∂ x L∂ x v G = 0, ∂ x Lv G = -∂ x v G , (1.13)
whereas the purely imaginary eigenvalues λ = ±iω n correspond to the eigenfunctions u = u ±n (x), which are smooth in x but decay algebraically as |x| → ∞.

Remark 1.5. Because the spectrum of ∂ x L is purely discrete by Theorem 1.4, we have no chance to assume dispersive decay estimates near the Gaussian solitary wave v G . This indicates that no asymptotic stability result can hold for solitary waves in the log-KdV equation (1.1).

Remark 1.6. Remark 1.5 agrees with the result of [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF]Proposition 4.3] stating that the L p norms at the solution v for any p 2 including p = ∞ may not vanish as t → ∞ (or in a finite time), hence the solution does not scatter to zero. Although this statement was proved for the log-NLS equation, it is based on the consideration of the same energy functional E(v) as in the log-KdV equation (1.1). This non-scattering result is related to the fact that the L p norm at the family (1.2) can be scaled to be arbitrarily small by using the scaling parameter c ∈ R.

Theorem 1.4 can be used to provide an alternative proof of Theorem 1.1. On the other hand, because of the algebraic decay of the eigenfunctions in Theorem 1.4, any function of x that decays like the Gaussian function as |x| → ∞ cannot be simply represented as series of eigenfunctions of the linearized operator ∂ x L. To explain the importance of such representations, we set v(x, t) := v G (x) + w(x, t) and obtain the equivalent log-KdV equation

w t = ∂ x Lw -∂ x N (w), (1.14)
where the nonlinear term N (w) is given by

N (w) := w log 1 + w v G + v G log 1 + w v G - w v G .
It is clear that the nonlinear term N (w) does not behave uniformly in x unless w decays at least as fast as v G in (1.3). On the other hand, if w(x, t) = v G (x)h(x, t), where h is a bounded function in its variables, then N (w) = v G n(h), where n(h) := h log(1 + h) + log(1 + h) -h is analytic in h for any h ∈ (-1, 1). This observation on the nonlinear term N (w) inspires us to consider solutions of the linearized log-KdV equation (1.7) starting with the initial data

u 0 ∈ X c such that u 0 = v G h 0 for h 0 ∈ L ∞ (R) ∩ L 2 (R).
We have undertaken numerical simulations of this linear Cauchy problem to illustrate that solutions u(x, t) of the linearized log-KdV equation (1.7) with Gaussian initial data do not spread out as the time variable evolves. Nevertheless, they produce visible radiation at the left slope of the Gaussian solutions. Further studies are needed to figure out if the nonlinear orbital stability of the Gaussian solitary wave v G can be proved in the framework of the nonlinear evolution problem (1.14) with initial data

w 0 = v G h 0 for small h 0 ∈ L ∞ (R) ∩ L 2 (R).
The rest of the paper consists of the following. Theorem 1.2 is proved in Section 2. Theorem 1.4 is proved in Section 3. Section 4 contains the proof of Theorem 1.1 (which is alternative to the one given in [START_REF] James | Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials[END_REF]) and the numerical simulations of the time evolution of the linearized log-KdV equation (1.7) starting with Gaussian initial data.

Global solutions of the log-KdV equation

To prove Theorem 1.2, we follow the algorithm developed in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] for the log-NLS equation (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Section 9.3]), which shares common features with the approach of Ginibre and Velo [START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF]. We slightly simplify the functional framework from [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF], by using Strichartz estimates instead of arguments coming from convex analysis. In the context of KdV equations, these tools were developed by Kenig, Ponce, and Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF].

The strategy of the proof is the following. We first regularize the logarithmic nonlinearity near the origin and construct a sequence of approximating solutions via a contraction principle as in [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF]. Then, we derive uniform estimates on the H 1 (R) norm of the approximating solutions. This allows us to pass to the limit and obtain a global solution v in the energy space X with nonincreasing v L 2 and energy E(v). In the last subsection, we establish uniqueness of the global solutions v of the log-KdV equation (1.1) with bounded ∂ x log |v| by using special properties of the logarithmic nonlinearity.

Approximating solutions

We shall regularize the behavior of the logarithmic nonlinearity f (v) = v log |v| near v = 0. Compared to the approximation considered in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF], we will work with a different (much simpler) approximation of the logarithmic nonlinearity.

For any fixed ε > 0, let us define the family of regularized nonlinearities in the form

f ε (v) = f (v), |v| ε, p ε (v), |v| ε, (2.1)
where p ε is an odd polynomial of degree 2m + 1 such that

∂ k v p ε (ε) = ∂ k v f (ε) for all 0 k m.
In this way, we construct f ε ∈ C m (R) for any m ∈ N. For instance, for m = 1, we can compute

p ε (v) = log(ε) - 1 2 v + 1 2ε 2 v 3 , (2.2)
which yields f ε (v) ∈ C 1 (R). For m = 2, we compute p ε (v) = log(ε) - 3 4 v + 1 ε 2 v 3 - 1 4ε 4 v 5 , (2.3)
which yields f ε (v) ∈ C 2 (R).
As pointed out in Remark 2.3 below, for the proof of Theorem 1.2, it is actually sufficient to consider the approximation

f ε (v) with m = 2.
Global behavior of the function f ε (v) is still determined by the logarithmic nonlinearity f (v). If m 1, the function f ε (v) is globally Lipschitz and for any fixed ε > 0 there exists a positive constant C ε such that

|f ε (v) -f ε (u)| C ε (|v| + |u|)|v -u|, for every v, u ∈ R. (2.4)
If m 2, we also have

|f ′ ε (v)v x -f ′ ε (u)u x | C ε ((|v| + |u|)|v x -u x | + (|v x | + |u x |)|v -u|) , (2.5)
for every v, u, v x , u x ∈ R, where C ε is another constant, which may change from one line to another line. Of course, we realize from examples (2.2) and (2.3) that C ε → ∞ as ε → 0. For a given initial data v 0 , we shall now consider a sequence of the approximating Cauchy problems associated with the generalized KdV equations

v ε t + v ε xxx + f ′ ε (v ε )v ε x = 0, t > 0, v ε | t=0 = v 0 , (2.6)
Using the linear estimates and the contraction principle from the work of Kenig, Ponce and Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF], we have the following local well-posedness result.

Theorem 2.1. Fix s > 3 4 and assume that f ε ∈ C 2 (R) satisfy the global Lipschitz estimates (2.4) and (2.5). For any v 0 ∈ H s (R), there exists a time T ( v 0 H s ) > 0 and a unique solution of the generalized KdV equation (2.6) satisfying

(1) v ε ∈ C([-T, T ], H s (R)), (2) v ε x ∈ L 4 ([-T, T ], L ∞ (R)), (3) D s x v ε x L ∞ x L 2 T < ∞, (4) v ε L 2 x L ∞ T < ∞, where f L p x L q T := R T -T |f (t, x)| q dt p/q dx 1/p .
Moreover, the solution v ε depends continuously on the initial data v 0 in H s (R) and

v(t) L 2 = v 0 L 2 , for every t ∈ [-T, T ]. (2.7)
If in addition m 2 and s 1, then the energy is conserved:

E ε (v ε (t)) = E ε (v 0 ), for every t ∈ [-T, T ], (2.8
)

where E ε (v) := 1 2 R (v x ) 2 dx - R W ε (v)dx, W ε (v) := v 0 f ε (v)dv.
(2.9)

Remark 2.2. Although Theorem 2.1 was proved for the KdV equation with

f (v) = v 2 in [13],
Duhamel's principle was used to write the Cauchy problem (2.6) in the integral form

v ε (0) = S(t)v 0 - t 0 S(t -t ′ ) f ′ ε (v ε (t ′ ))v ε x (t ′ ) dt ′ , (2.10)
where S(t) is the solution operator associated with the group e -t∂ 3

x . Contraction principle is now applied in the same spirit as in [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] provided the regularized nonlinearity f ε satisfies the global Lipschitz estimates (2.4) and (2.5).

Remark 2.3. The solution is extended globally for any integer s ∈ N thanks to the a priori energy estimates provided that the nonlinearity f ε is sufficiently smooth. We only use this construction for s = 1, when f ε ∈ C 2 (R) is sufficient. In this context, the approximation (2.1) with (2.3) can be used for the rest of this work.

Uniform energy estimates

We work with local solutions of Theorem 2.1 for s = 1. Besides conservation of the L 2 norm in (2.7), the generalized KdV equation (2.6) admits conservation of the energy in (2.8), as it is stated in Theorem 2.1.

For the purpose of construction of global solutions in H 1 (R), we need to bound E ε (v) from below by the squared H 1 (R) norm. To do so, we only need to bound positive values of W ε (v) from above.

Since

f ′ (v) = log |v| + 1 → -∞ as v → 0, we realize that f ′ ε (v) = p ′ ε (v) ≪ -1 for |v| ε if ε > 0 is sufficiently small. Indeed, it follows from the explicit example (2.3) that f ′ ε (v) = p ′ ε (v) = log(ε) + O(1) for all |v| ε as ε → 0.
Therefore, if ε > 0 is sufficiently small, we have

W ε (v) = 1 2 [log(ε) + O(1)] v 2 0, |v| ε. (2.11) Let us define W (v) := 1 2 v 2 log |v| - 1 4 v 2 , ( 2
W ε (v) = W (v) + C m ε 2 , |v| ε. (2.13)
For instance, integration of the explicit example (2.3) yields C 2 = 1 12 . It follows from (2.11), (2.12), and (2.13) that there exists a positive ε-independent constant C such that

[W ε (v)] + [W (v)] + + C m v 2 C|v| 3 , for every v ∈ R.
(2.14) By Sobolev's embedding of H 1 (R) into L ∞ (R), we obtain from (2.9) and (2.14)

E ε (v) 1 2 v x 2 L 2 -C v L ∞ v 2 L 2 1 2 v x 2 L 2 -C v H 1 P (v),
where

P (v) := 1 2 v 2 L 2 . Because E ε (v ε
) and P (v ǫ ) are constants of motion for the solution v ǫ of the generalized KdV equation (2.6) in Theorem 2.1 with s = 1, we obtain the time-uniform bound on the H 1 norm of v ε :

v ε (t) H 1 CP (v 0 ) + 2E ε (v 0 ) + 2P (v 0 ) + C 2 P 2 (v 0 ) < ∞, t ∈ [-T, T ].
(2.15)

By using bound (2.15), we use a standard continuation argument for solutions of the integral equation (2.10) obtained via a contraction principle and continue the local solution

v ε ∈ C([-T, T ], H 1 (R)) to a global solution v ε ∈ C(R, H 1 (R))
. The global solution satisfies the timeuniform bound (2.15) extended for every t ∈ R.

Additionally, because of (2.14), there exists an ε-independent constant C such that

[W ε (v ε (t))] + L 1 C v ε (t) L ∞ v ε (t) 2 L 2 CP (v 0 ) CP (v 0 ) + 2E ε (v 0 ) + 2P (v 0 ) + C 2 P 2 (v 0 ) . (2.16)
For the negative part of W ε , we obtain

[W ε (v ε (t))] -L 1 = [W ε (v ε (t))] + L 1 - R W ε (v ε (t))dx = [W ε (v ε (t))] + L 1 + E ε (v ε (t)) - 1 2 v ε x 2 L 2 CP (v 0 ) CP (v 0 ) + 2E ε (v 0 ) + 2P (v 0 ) + C 2 P 2 (v 0 ) + E ε (v 0 ).(2.17)
Bounds (2.16) and (2.17) allow us to control W ε (v ε (t)) L 1 for every t ∈ R.

Passage to the limit

We shall now consider the limit ε → 0 for the sequence of global approximating solutions v ε ∈ C(R, H 1 (R)) satisfying the generalized KdV equations (2.6). We recall definition (1.9) of the energy space X for the log-KdV equation (1.1). Assume that v 0 ∈ X so that

E(v 0 ) < ∞. Clearly E ε (v 0 ) < ∞ for any v 0 ∈ H 1 (R).
Since f ε (0) = f (0) = 0, we have the pointwise limit f ε (v) → f (v) as ε → 0 for every v ∈ R for the sequence of regularized nonlinearities in (2.1). Consequently, we have the pointwise limit of W ε (v) to the potential W (v) given by (2.12):

W ε (v) → W (v) as ε → 0 for every v ∈ R.
In view of (2.3), there exists a positive ε-independent constant C such that

W ε (v) - 1 2 (log ε)v 2 Cv 2 , |v| ε.
We infer from the explicit expression of W given by (2.12) that

|W ε (v)| |W (v)| + Cv 2 , |v| ε.
On the other hand, a similar estimate holds in the region |v| ε, thanks to the relation (2.13), so there exists a positive ε-independent constant C such that

|W ε (v)| |W (v)| + Cv 2 , for every v ∈ R.
By Lebesgue's dominated convergence theorem, we have

R W ε (v)dx → R W (v)dx as ε → 0, for every v ∈ X.
Therefore, E ε (v 0 ) → E(v 0 ) as ε → 0. By using the above estimates and the bounds (2.15), (2.16), and (2.17), we obtain the following ε-and t-uniform bound for the sequence of approximating solutions

v ε ∈ C(R, H 1 (R)) starting with the initial data v 0 ∈ X: v ε (t) H 1 + W ε (v ε (t)) L 1 C(P (v 0 ), E(v 0 )) < ∞, ∀t ∈ R, ∀ε ∈ (0, 1], (2.18)
where the positive constant C depends on the initial values P (v 0 ) and E(v 0 ) only. Therefore, the sequence of approximating solutions v ε is bounded in space L ∞ (R, X). It also follows from the generalized KdV equation (2.6) that the sequence

v ε t is bounded in space L ∞ (R, H -2 (R)). From Arzela-Ascoli Theorem, there exist v ∈ L ∞ (R, H 1 (R)) and a subsequence of v ε ∈ L ∞ (R, H 1 (R)), still denoted by v ε , such that v ε → v strongly in L ∞ loc (R, H s loc (R)) as ε → 0, for all s < 1. (2.19)
Because the limit (2.19) includes the range 1 2 , 1 for s, up to subtracting another subsequence, we may assume that v ε (x, t) → v(x, t) as ε → 0, for almost every x ∈ R and for every t ∈ R.

(2.20)

In addition, because the upper bound (2.18) also controls W ε (v ε (t)) L 1 , Fatou's lemma shows that the limiting function v belongs to L ∞ (R, X).

To show that P (v(t)) P (v 0 ) and E(v(t)) E(v 0 ) for every t ∈ R, we use the weak lower semicontinuity of the H 1 norm and Fatou's lemma for the potential term [W ε ] -to obtain

v(t) 2 H 1 lim ε→0 v ε (t) 2 H 1 , [W ε (v(t))] -L 1 lim ε→0 [W ε (v ε (t))] -L 1 , for every t ∈ R.
On the other hand, using the relation (2.13), the limit (2.20), and the fact that W (v) 0 for |v| √ e, we obtain

[W (v(t))] + L 1 = lim ε→0 [W ε (v ε (t))] + L 1 , for every t ∈ R.
Then, it follows from the conservation

P (v ε (t)) = P (v 0 ) and E ε (v ε (t)) = E ε (v 0 ) for every t ∈ R and the convergence E ε (v 0 ) → E(v 0 ) as ε → 0 that P (v(t)) P (v 0 ) and E(v(t)) E(v 0 )
It remains to show that the limiting function v ∈ L ∞ (R, X) is a weak solution of the log-KdV equation (1.1). To do so, we first write a weak formulation of the generalized KdV equation

(2.6). If v ε ∈ C(R, H 1 (R)) is a solution in Theorem 2.1 for T = ∞, then for any test functions ψ ∈ C ∞ 0 (R x ) and ϕ ∈ C ∞ 0 (R t ), we have R v ε , ψ L 2 ϕ ′ (t) + v ε , ψ ′′′ L 2 ϕ(t) dt + R R f ε (v ε )ψ ′ (x)ϕ(t)dxdt = 0. (2.21) Since f ε (v) → f (v) pointwise in v as ε → 0,
v, R v, ψ L 2 ϕ ′ (t) + v, ψ ′′′ L 2 ϕ(t) dt + R R f (v)ψ ′ (x)ϕ(t)dxdt = 0, (2.22)
where ψ and ϕ are any test functions. Therefore, v ∈ L ∞ (R, X) is a weak solution of the log-KdV equation (1.1), in particular, v t ∈ L ∞ (R, H -2 (R)). The existence part of Theorem 1.2 is now proven.

Remark 2.4. Since v t is in H -2 (R) (as opposed to the NLS case, where it is defined in H -1 (R), see [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF]), we are not able to establish even conservation of the L 2 norm for the weak solutions of the log-KdV equation (1.1), since we cannot prove rigorously that

d dt v(t) 2 L 2 = 2 v t , v L 2 = 0.
Nevertheless, we have proved that v(t) 2 L 2 is a non-increasing function of time t.

Uniqueness of solutions

We will show uniqueness of the solution v ∈ L ∞ (R, X) under the additional condition

(log |v|) x ∈ L ∞ ((-t 0 , t 0 ) × R).
Provided that uniqueness is proven, continuous dependence on the initial data v 0 ∈ X, the L 2 norm and energy conservation, and continuity of the solution v ∈ C((-t 0 , t 0 ), X) in time t follow from the arguments identical to the case of the log-NLS equation [5, Section 9.3]. Assume existence of two local solutions v and u of the log-KdV equation (1.1) starting with the same initial data v 0 . Set w := v -u such that w| t=0 = 0. Then w satisfies a weak formulation similar to the integral equation (2.22) for the partial differential equation

w t + w xxx + (v log |v| -u log |u|) x = 0.
Multiplying this equation by w, integrating over x, and formally neglecting the values of w and its derivatives as |x| → ∞, we obtain

d dt P (w) = - R (v x log |v| -u x log |u|)wdx,
where P (w) = 1 2 w 2 L 2 . We use the following bound for the log-nonlinearity [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]:

| log |v| -log |u|| |v -u| min(|v|, |u|) . ( 2

.23)

We then write

d dt P (w) = - |v|<|u| (v x log |v| -u x log |u|)wdx - |v| |u| (v x log |v| -u x log |u|)wdx = - |v|<|u| v x (
u x (log |v| -log |u|)wdx v x v L ∞ + u x u L ∞ 2P (w).
Integrating by parts, we also have

- |v|<|u| log |u|ww x dx - |v| |u| log |v|ww x dx = 1 2 |v|<|u| (log |u|) x w 2 dx + 1 2 |v| |u| (log |v|) x w 2 dx, hence d dt P (w) 3 v x v L ∞ + u x u L ∞ P (w).
Gronwall's inequality implies that if w| t=0 = 0, then P (w) = 0 for all t ∈ (-t 0 , t 0 ), for which the solutions v, u of the log-KdV equation (1.1) satisfy (log |v|) x , (log |u|) x ∈ L ∞ ((-t 0 , t 0 ) × R). The uniqueness part of Theorem 1.2 is now proven.

Spectral stability of Gaussian solitary waves

To prove Theorem 1.4, we study the spectrum of the linearized operator A := ∂ x L given by

A = -∂ 3 x + 1 4 (x 2 -6)∂ x + 1 2 x. (3.1)
The domain of this operator with a range in L 2 (R) is given by

D(A) = u ∈ H 3 (R) : x 2 ∂ x u ∈ L 2 (R), xu ∈ L 2 (R) . (3.2)
We note that the eigenfunctions of the linearized operator ∂ x L associated with the KdV-type spectral problem Af = λf is defined in the function space X A := D(A) ∩ Ḣ-1 (R), hence the anti-derivative of the eigenfunction f is required to be squared integrable [START_REF] Kapitula | A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems[END_REF][START_REF] Pelinovsky | Spectral stability of nonlinear waves in KdV-type evolution equations, Spectral analysis, stability, and bifurcation in modern nonlinear physical systems[END_REF]. We shall employ the Fourier transform F :

L 2 (R) → L 2 (R) defined by û(k) := F(u)(k) = 1 √ 2π R u(x)e -ikx dx, k ∈ R. (3.3)
The Fourier transform F is helpful in the study of the spectrum of A because of dualism between derivatives and multiplications. After an application of the Fourier transform F, the third-order differential operator A in x-space is mapped to the second-order differential operator  in k-space, where

 = i 4 k -∂ 2 k + 4k 2 -6 . (3.4)
The function space X A = D(A) ∩ Ḣ-1 (R) is now mapped to the function space XA given by

XA = û ∈ H 1 (R) : k∂ 2 k û ∈ L 2 (R), k 3 û ∈ L 2 (R), k -1 û ∈ L 2 (R) . (3.5) Denoting B := k(-∂ 2 k + 4k 2 -6), we have σ( Â) = i 4 σ( B).

Double zero eigenvalue of Â

Let vG (k) := e -k 2 be the Fourier transform of the Gaussian solitary wave v G (up to the constant multiplicative factor). We check by direct computation that

B∂ k vG = 0 and Bv G = -4kv G = 2∂ k vG . (3.6)
Since B = 4k L, where L is the Fourier image of the Schrödinger operator L with a harmonic potential (1.5), we conclude that null( B) = span{∂ k vG }, whereas the generalized null space of B includes the two-dimensional subspace X0 = span{∂ k vG , vG }.

Moreover, since B * vG = 0 and vG L 2 = 0, no solution û ∈ XA of the inhomogeneous equation B û = vG exists. Therefore, the Jordan chain for the zero eigenvalue of B is two-dimensional and the generalized null space of B is exactly X0 . With the inverse Fourier transform, these arguments conclude consideration of the zero eigenvalue in Theorem 1.4.

Nonzero eigenvalues of Â

Let us now consider nonzero values for the spectral parameter λ in the spectral problem  f = λ f . Recall that σ( Â) = i 4 σ( B). Therefore, let λ = i 4 E and consider the spectral problem B û = E û rewritten as the following differential equation

d 2 û dk 2 + E k + 6 -4k 2 û(k) = 0, k ∈ R. (3.7)
We employ the theory of differential equations to study solutions of the spectral problem (3.7).

If E = 0, the point k = 0 is a regular singular point of the differential equation (3.7) with indices 0 and 1. By Frobenius' method [START_REF] Teschl | Ordinary differential equations and dynamical systems[END_REF]Chapter 4], there exist two linearly independent solutions of this differential equation. The first solution is given by the power series expansion

û1 (k) = k - 1 2 Ek 2 + 1 12 E 2 -1 k 3 + O(k 4 ) as k → 0 (3.8)
and the other solution is given by the logarithm-modified power series expansion

û2 (k) = 1 -3 1 + 1 4 E 2 k 2 + O(k 3 ) -Ek log(k) 1 - 1 2 Ek + O(k 2 ) as k → 0. (3.9)
Because the other singular point of the differential equation (3.7) is infinity, the expansions (3.8) and (3.9) converge for every k ∈ R.

Recall again the function space XA defined by (3.5). Because k -1 û ∈ L 2 (R) is required in Xk , we have û ∈ XA if and only if û is constant proportional to the solution û1 . Hence, both for k > 0 and k < 0, the solution of the differential equation (3.7) is uniquely defined by the solution û1 (up to the constant multiplicative factor) for every E ∈ C\{0}. The admissible values of E are determined from the behavior of the solution û(k) as |k| → ∞.

By the WKB method without turning points [14, Chapter 7.2], there exist two linearly independent solutions of the differential equation (3.7): one solution diverges and the other one decays to zero as |k| → ∞. The decaying solution is defined by the asymptotic behavior

û(k) ∼ ke -k 2 1 - E 4k + O 1 k 2 as |k| → ∞. (3.10)
It becomes now clear that two decay conditions at k → +∞ and k → -∞ over-determine the spectral problem (3.7) because the behavior of û both for k > 0 and k < 0 is uniquely determined by only one spectral parameter E (up to the constant multiplicative factors) and no parity symmetry exists for k > 0 and k < 0. The way around this obstacle is to consider a weak formulation for the solutions of the differential equation (3.7) in the function space XA . Since the differential equation with E = 0 has singularities at {-∞, 0, ∞}, we split R into two sets I -:= (-∞, 0) and I + := (0, ∞) and look for piecewisely defined eigenfunctions û± supported on I ± only.

Let B± denote the operator B restricted on I ± subject to the Dirichlet boundary condition at k = 0. By Theorem 4 in [8, p.1438], the continuous spectrum of B is the union of the continuous spectra of operators B + and B -.

We shall first characterize the spectrum of the operator B+ on I + and prove that this spectrum is purely discrete. By using a substitution û+ (k) = k 1/2 v+ (k) for k > 0, we reduce the spectral problem B+ û+ = E û+ to the symmetric form

k 1/2 - d 2 dk 2 + 4k 2 -6 k 1/2 v+ (k) = Ev + (k), k ∈ I + . (3.11)
Since k = 0 is singular, we require |v + (0)| < ∞, although the solution û1 in (3.8) implies more precisely that v+ (k) = O(k 1/2 ) as k → 0. The spectral problem (3.11) in the function space with |v + (0)| < ∞ is self-adjoint, hence the admissible values of E are real.

We shall now claim that σ( B+ ) = {E n } n∈N + , where 0 = E 0 < E 1 < E 2 < ... and E n → +∞ as n → ∞. First, because the potential 4k 2 -6 is confining, the resolvent operator is compact, and the spectrum of B+ is purely discrete. Next, v+ = k 1/2 e -k 2 is the exact solution of the spectral problem (3.11) for E = 0 and this eigenfunction is strictly positive for all k ∈ I + . By Sturm's Theorem, E 0 = 0 is at the bottom of the spectrum of B+ , which is bounded from below, and hence E n > 0 for every n ∈ N. Finally, the eigenfunctions are uniquely determined by the expansion (3.8) as k → 0, hence each eigenvalue is simple.

As a consequence of the reduction to the self-adjoint problem (3.11), we have orthogonality and normalization of the real-valued eigenfunctions û+n (k) = k 1/2 v+n (k) corresponding to the real eigenvalues E n for all n ∈ N + :

δ n,m = ∞ 0 v+n (k)v +m (k)dk = ∞ 0 û+n (k)û +m (k) k dk, (3.12)
where δ n,m is the Kronecker's symbol.

The same analysis applies to the spectrum of the operator Bon I -with the only difference that the substitution û-

(k) = (-k) 1/2 v-(k) for k < 0 reduces the spectral problem B-û-= E û- to the symmetric form (-k) 1/2 - d 2 dk 2 + 4k 2 -6 (-k) 1/2 v-(k) = -Ev -(k), k ∈ I -, (3.13)
subject to the boundary behavior |v

-(0)| < ∞. Therefore, σ( B-) = {-E n } n∈N +
, where E n are the same eigenvalues as in the spectral problem (3.11). Similarly to (3.12), it follows from the selfadjoint problem (3.13) that the real-valued eigenfunctions û-n (k) = (-k) 1/2 v-n (k) corresponding to the real eigenvalues -E n for all n ∈ N + are orthogonal and can be normalized by

δ n,m = 0 -∞ v-n (k)v -m (k)dk = - 0 -∞ û-n (k)û -m (k) k dk. (3.14)
The statement about nonzero eigenvalues of the operator A in Theorem 1.4 is now proven with the correspondence ω n = 1 4 E n for all n ∈ N. Remark 3.1. The double zero eigenvalue associated with the two-dimensional subspace X0 = span{∂ k vG , vG } for the operator  is mapped into a semi-simple zero eigenvalue associated with the simple zero eigenvalues of the operators B+ and B-. There is no ambiguity here because if E = 0, then k = 0 is an ordinary point of the differential equation (3.7) and the splitting of R into I -and I + does not make sense.

Eigenfunctions of the spectral problem

We now confirm that the eigenfunction û defined piecewise as

either û(k) = û+ (k), k > 0, 0, k < 0, or û(k) = 0, k > 0, û-(k), k < 0,
belongs to the space XA . We note that û is continuous on R and piecewise C 1 with the jump discontinuity of the first derivative at k = 0. Moreover, û(k) = O(k) as k → 0. Therefore, û ∈ H 1 (R) and k -1 û ∈ L 2 (R). Furthermore, û(k) decays to zero as |k| → ∞ according to the asymptotic behavior (3.10). Therefore, k 3 û ∈ L 2 (R). Finally, ∂ 2 k û is proportional to the Dirac delta distribution δ at k = 0 and is smooth for k = 0, hence

∂ 2 k û / ∈ L 2 (R) but k∂ 2 k û ∈ L 2 (R). Thus, û ∈ XA .
The Fourier transform F is dual between the smoothness and decay of the eigenfunctions. Since the eigenfunctions û(k) decays fast as |k| → ∞, according to the Gaussian decay (3.10), their inverse Fourier transform u(x) is smooth in x. Moreover, u ± defined by

u + (x) = 1 √ 2π ∞ 0 û+ (k)e ikx dk, u -(x) = 1 √ 2π 0 -∞ û-(k)e ikx dk (3.15)
are analytically extended for Im(x) ≷ 0 respectively. On the other hand, û(k) have jump discontinuity in the first derivative across k = 0, therefore, u(x) decay only algebraically as |x| → ∞. The only exception is the double zero eigenvalue, since one can glue û+ (k) and û-(k) for the eigenvalue E 0 = 0 to a smooth eigenfunction û(k) = ke -k 2 , which corresponds to the eigenfunction u(x) = xe -x 2 /4 with the fast decay as |x| → ∞.

The statement about eigenfunctions of the operator A in Theorem 1.4 is now proven.

Numerical illustrations

= E 0 < E 1 < E 2 .
Eigenfunctions for E 1 and E 2 have one and two zeros for k > 0, respectively.

Figure 2 shows the real (left) and imaginary (right) parts of the eigenfunctions u + versus x after the inverse Fourier transform (3.15). The imaginary part of the first eigenfunction decays fast as |x| → ∞, according to the exact eigenfunction v ′ G (x) ∼ xe -x 2 /4 , which is also shown by a dotted line (invisible from the numerical dots).

To inspect the slow (algebraic) decay of the eigenfunctions, we multiply the eigenfunctions by the factor (1 + x 2 ) p/2 with p = 2 and p = 3. It follows from the numerical data that the real parts of the eigenfunctions decay like O(|x| -2 ) as |x| → ∞, whereas the imaginary parts of the eigenfunctions (except for the first eigenfunction) decay like O(|x| -3 ) as |x| → ∞. This corresponds to the finite jump discontinuity of the first derivative of û(k) at k = 0.

Cauchy problem for the linearized log-KdV equation

We consider the Cauchy problem for the linearized log-KdV equation (1.7) at the Gaussian solitary wave v G . For convenience, we rewrite the Cauchy problem again:

u t = ∂ x Lu, t > 0, u| t=0 = u 0 , (4.1)
We shall first make use of Theorem 1.4 to give a quick proof of Theorem 1.1. This proof is alternative to the arguments given in [START_REF] James | Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials[END_REF], which rely on the symplectic projections and the energy method. Next, we shall approximate the Cauchy problem numerically, starting with the Gaussian initial data. We conclude the paper with a short discussion.

Proof of Theorem 1.1

Let {u ±n (x)} n∈N be a sequence of eigenfunctions of ∂ x L for the sequence of nonzero imaginary eigenvalues {±iω n } n∈N constructed in Theorem 1.4. It follows from the orthogonality and normalization conditions (3.12) and (3.14) that the eigenfunctions are orthogonal to each other and normalized with respect to the symplectic inner product:

∂ -1 x u ±n , u ±m L 2 := ∞ -∞ û±n (k)û ±m (k) ik dk = ∓iδ n,m , (4.2)
where

∂ -1 x u(x) := x -∞ u(x ′ )dx ′ and the Fourier transform (3.3) is used. Note that u ±n ∈ L 1 (R) if u ±n ∈ L 2 1 (R) ⊂ D(A). Also note that û±n (0) = 0 if u ±n ∈ Ḣ-1 (R) ⊂ X A .
Let us consider the decomposition of the solution u of the linearized log-KdV equation (4.1) as the series of eigenfunctions of the operator A:

u(x, t) = b v G (x) -tv ′ G (x) + a 0 v ′ G (x) + n∈N a +n u +n (x)e iωnt + n∈N a -n u -n (x)e -iωnt , (4.3)
where the coefficients b and {a n } n∈Z are found uniquely from the conditions of symplectic orthogonality and normalization (4.2) applied to the initial data u| t=0 = u 0 :

b = v G , u 0 L 2 v G , v G L 2 , a 0 = 1 v G , v G L 2 b 2 ∞ -∞ v G (x)dx 2 -∂ -1 x v G , u 0 L 2 , (4.4)
and

a ±n = ±i ∂ -1 x u ±n , u 0 L 2 , n ∈ N. (4.5)
If u 0 ∈ X c defined in (1.6), then b = 0. The coefficients {a n } n∈Z are well defined for u 0 ∈ X c . Moreover, by the spectral theorem and the reduction of the non-self-adjoint operator A to the two self-adjoint operators in Section 3.2, the series (4.3) converges in the L 2 sense for every t ∈ R. Define now the conserved energy of the linearized log-KdV equation,

E c (u) := 1 2 Lu, u L 2 = const, t ∈ R. (4.6) Recall that E c (u 0 ) 0 if u 0 ∈ X c . It follows from (4.3) and (4.6) that if u 0 ∈ X c , then u(•, t) ∈ X c and E c (u(•, t)) 0 for every t ∈ R. Let y(x, t) := u(x, t) -a 0 v ′ G (x)
and recall that b = 0 for u 0 ∈ X c . Under these conditions, we have

E c (y(•, t)) = E c (u(•, t)) = E c (u 0 ), t ∈ R and E c (y(•, t)) = 1 2 n∈N ω n |a +,n | 2 + |a -n | 2 ,
where the orthogonality and normalization conditions (4.2) have been used. Because y belongs to a subspace of X c spanned by {u ±n } n∈N associated with the nonzero eigenvalues {±iω n } n∈N , where each ω n > 0, the energy functional E c (y) is equivalent to the squared H 1 (R) ∩ L 2 1 (R) norm. Therefore, there is a t-independent positive constant C such that

y(•, t) 2 H 1 + y(•, t) 2 L 2 1 CE c (y(•, t)) = CE c (u 0 ) C 1 2 u 0 2 H 1 + 1 8 u 0 2 L 2 1 . (4.7)
By the decomposition (4.3), the bound (4.7) and the triangle inequality, we obtain the bound (1.8) for every t ∈ R. Theorem 1.1 is now proven.

Numerical illustrations

We truncate the spatial domain on [-L, L] for sufficiently large L (L = 40 was used in our numerical results) and discretize u(x, t) at equally spaced grid points {x k } N +1 k=0 subject to the periodic boundary conditions u(x 0 , t) = u(x N +1 , t). Using the central difference approximation and the Heun's method, we rewrite the linearized log-KdV equation (1.7) in the iterative form

I - ∆t 2 DL D u m+1 = I + ∆t 2 DL D u m (4.8)
where ∆t is the time step, u m is the vector of discretized solution at the time t m , I is an identity matrix, D is the matrix for the first derivative, and L D is the matrix for the second-order differential operator L.

For a positive parameter α, we consider the odd initial data

u 0 (x) = x x 2 - 5 + 6α (1 + α)(1 + 2α) e -(1+2α)x 2 4 , (4.9)
which satisfies the symplectic orthogonality conditions v G , u 0 L 2 = ∂ -1

x v G , u 0 L 2 = 0. These constraints ensure that b = 0 and a 0 = 0 in the decomposition (4.3) and the solution u of the linearized log-KdV equation (4.1) is spanned by eigenfunctions of the linearized operator A = ∂ x L for nonzero eigenvalues.

Figure 3 reports numerical computations of the iterative scheme (4.8) for the Gaussian initial data (4.9) with α = 0.1. Besides the profiles of the solution shown on the spatial interval [-15, 15] (bottom panel), we also show the center of mass (top left panel) .

x(t) = R xu 2 (x,
We can see from the top right panel of Figure 3 that the standard deviation σ(t) oscillates periodically with a large amplitude, which is still much smaller than the half-size of the computational domain L = 40. Therefore, the odd Gaussian data (4.9) evolves into a solution, which does not spread out in the time evolution of the linearized log-KdV equation (1.7). Nevertheless, a visible radiation appears on the left slope of the Gaussian pulse.

Discussion

The numerical results of Section 4.2 support the conjecture that if the initial data u 0 ∈ X c of the linearized log-KdV equation (1.7) satisfies the constraint

u 0 (x) = v G (x)h 0 (x) with h 0 ∈ L ∞ (R) ∩ L 2 (R),
then there exists a unique solution of the linearized log-KdV equation (1.7) in the form u(x, t) = v G (x)h(x, t), where h(•, t) ∈ L ∞ (R) ∩ L 2 (R) for every t ∈ R and h(•, 0) = h 0 .

If this result can be established rigorously, then one can analyze the linearized log-KdV equation with a source term to complete analysis of the equivalent log-KdV equation (1.14) with the initial data w 0 = v G h 0 , where h 0 ∈ L ∞ (R) ∩ L 2 (R) and h 0 L ∞ ∩L 2 is sufficiently small. This route may lead to the proof of orbital stability of the Gaussian solitary wave v G for the log-KdV equation (1.1) in the class of functions with initial data v 0 = (1+h 0 )v G , where h 0 ∈ L ∞ (R)∩L 2 (R) and h 0 L ∞ ∩L 2 is sufficiently small. However, this work is still to be done, hence the problem of nonlinear orbital stability of the Gaussian solitary wave v G remains opened for further studies.

1 2 -x 2 4

 12 , x ∈ R. (1.3) 1

  .12) and denote the positive part of W by [W ] + . Note that W (v) 0 for |v| √ e. Because f ε (v) = f (v) for |v| ε, there exists a positive constant C m , which only depends on m such that

Figure 1

 1 Figure1shows the first three eigenfunctions û+ (k) = k 1/2 v+ (k) of the spectral problem(3.11) for the first three eigenvalues E 0 = 0, E 1 ≈ 5.4109, and E 2 ≈ 12.3080. These eigenfunctions and eigenvalues were computed numerically by means of the central difference approximation of the derivatives in the spectral problem(3.11) and the MATLAB eigenvalue solver. The fast decay of the eigenfunctions û+ (k) as k → ∞ and the Sturm's nodal properties of the eigenfunctions for k > 0 are obvious from the figure.

Figure 1 :

 1 Figure 1: Eigenfunctions û of the spectral problem (3.7) versus k for the first three eigenvalues 0= E 0 < E 1 < E 2 .Eigenfunctions for E 1 and E 2 have one and two zeros for k > 0, respectively.

Figure 2 :

 2 Figure 2: Real (left) and imaginary (right) parts of the three eigenfunctions u versus x for the first three eigenvalues E.

Figure 3 :

 3 Figure 3: Numerical solution of the linearized log-KdV equation (1.7) with an odd initial data (4.9): center of mass x (top left) and standard deviation σ (top right) versus time; the profile u(x, t) versus x (bottom) for t = 0 (blue dashed), t = 2.5 (green dotted), and t = 5 (red solid).

Figure 4

 4 Figure 4 reports numerical computations for the even initial datau 0 (x) = x 4 -3(3 + 4α) (1 + α)(1 + 2α) x 2 + 6 (1 + α)(1 + 2α) e -(1+2α)x 2 4

Figure 4 :

 4 Figure 4: Numerical solution of the linearized log-KdV equation (1.7) with the even initial data (4.10): center of mass x (top left) and standard deviation σ (top right) versus time; the profile u(x, t) versus x (bottom) for t = 0 (blue dashed), t = 2.5 (green dotted), and t = 5 (red solid).

Acknowledgement. D.P. is indebted to Guillaume James (University of Grenoble) for bringing up the problem and fruitful collaboration, as well as to Thierry Gallay (University of Grenoble) who noticed the visible radiation in our numerical simulations. D.P. is supported by the CNRS Visiting Fellowship. He thanks Institut de Mathématiques et de Modélisation, Université Montpellier for hospitality and CNRS for support during his visit (September-November, 2013).


S Fournais

N Raymond

OPTIMAL MAGNETIC SOBOLEV CONSTANTS IN THE SEMICLASSICAL LIMIT

This paper is devoted to the semiclassical analysis of the best constants in the magnetic Sobolev embeddings in the case of a bounded domain of the plane carrying Dirichlet conditions. We provide quantitative estimates of these constants (with an explicit dependence on the semiclassical parameter) and analyze the exponential localization in L ∞ -norm of the corresponding minimizers near the magnetic wells.

Preliminary considerations and main results

We are interested in the following minimization problem. For the sake of simplicity, we consider a simply connected bounded domain Ω ⊂ R 2 , p ∈ [2, +∞), h > 0 and a smooth vector potential A on Ω. We introduce the following "nonlinear eigenvalue" (or optimal magnetic Sobolev constant):

(1. [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF] λ(Ω, A, p, h) = inf

ψ∈H 1 0 (Ω),ψ =0 Q h,A (ψ)
Ω |ψ| p dx 2 p = inf ψ∈H 1 0 (Ω), ψ L p (Ω) =1 Q h,A (ψ),
where the magnetic quadratic form is defined by

∀ψ ∈ H 1 0 (Ω), Q h,A (ψ) = Ω |(-ih∇ + A)ψ| 2 dx.
The Dirichlet realization of the magnetic Laplacian on Ω (defined as the Friedrichs extension) is denoted by L h,A whose domain is

Dom (L h,A ) = {ψ ∈ Dom(Q h,A ) = H 1 0 (Ω) : (-ih∇ + A) 2 ψ ∈ L 2
(Ω)}. We recall that B = ∇ × A is called the magnetic field (with the notation A

= (A 1 , A 2 ), B = ∂ 2 A 1 -∂ 1 A 2 ).
Let us here notice that there exists a vast literature dealing with the case p = 2. In this case λ(Ω, A, 2, h) is the lowest eigenvalue of the magnetic Laplacian. On this subject, the reader may consult the books and reviews [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Helffer | Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field[END_REF][START_REF] Raymond | Little Magnetic Book[END_REF].

1.1. Motivations and context. Before describing the motivations of this paper, let us recall some basic facts concerning the minimization problem (1.1).

Lemma 1.1. The infimum in (1.1) is attained.

Proof. The proof is standard but we recall it for completeness. Consider a minimizing sequence (ψ j ) that is normalized in L p -norm. Then, by a Hölder inequality and using that Ω has bounded measure, (ψ j ) is bounded in L 2 . Since A ∈ L ∞ (Ω), we conclude that (ψ j ) is bounded in H 1 0 (Ω). By the Banach-Alaoglu Theorem there exists a subsequence (still denoted by (ψ j )) and ψ ∞ ∈ H 1 0 (Ω) such that ψ j ψ ∞ Date: November 20, 2014.

weakly in H 1 0 (Ω) and ψ j → ψ ∞ in L q (Ω) for all q ∈ [2, +∞). This is enough to conclude.

Let us now consider the (focusing) equation satisfied by the minimizers.

Lemma 1.2. The minimizers (which belong to H 1 0 (Ω)) of the L p -normalized version of (1.1) satisfy the following equation in the sense of distributions:

(1.2) (-ih∇ + A) 2 ψ = λ(Ω, A, p, h)|ψ| p-2 ψ, ψ L p (Ω) = 1.

In particular (by Sobolev embedding), the minimizers belong to the domain of L h,A .

This paper is motivated by the seminal paper [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF] where the minimization problem (1.1) is investigated for Ω = R d and with a constant magnetic field (and also in the case of some nicely varying magnetic fields). In particular, Esteban and Lions prove the existence of minimizers by using the famous concentration-compactness method. In the present paper, we want to describe the minimizers as well as the infimum itself in the semiclassical limit h → 0. The naive idea is that, locally, modulo a blow up argument, they should look like the minimizers in the whole plane. In our paper, we will also allow the magnetic field to vanish and this will lead to other minimization problems in the whole plane which are interesting in themselves and for which the results of [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF] do not apply.

Another motivation to consider the minimization problem (1.1) comes from the recent paper [START_REF] Di Cosmo | Semiclassical stationary states for nonlinear Schrödinger equations with a strong external magnetic field[END_REF]. Di Cosmo and Van Schaftingen analyze a close version of (1.1) in Ω ⊂ R d in the presence of an additional electric potential. Note here that, as for the semiclassical analysis in the case p = 2, if there is a non-zero electric potential, then the minima of V tend to attract the bound states, independently from the presence of the magnetic field [START_REF] Kurata | Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields[END_REF]; in [START_REF] Di Cosmo | Semiclassical stationary states for nonlinear Schrödinger equations with a strong external magnetic field[END_REF] the electric potential is multiplied by h and plays on the same scale as B which cannot be treated as a perturbation. These authors prove, modulo subsequences extractions of the semiclassical parameter, that the asymptotics of the optimal Sobolev constant (with electro-magnetic field) is governed by a family of model minimization problems with constant magnetic fields and electric potentials. They also establish that we can find minimizers of (1.1) which are localized near the minima of the "concentration function". This function is nothing but the infimum of the model problem in R d , depending on the point x where the blow up occurs. In all their estimates, these authors do not quantify the convergence with an explicit dependence on h.

In our paper we will especially tackle this question in dimension two in the case of a pure magnetic field and we will see how this refinement can be applied to get localization estimates. As in [START_REF] Di Cosmo | Semiclassical stationary states for nonlinear Schrödinger equations with a strong external magnetic field[END_REF], we will first give upper bounds of λ(Ω, A, p, h), but with quantitative remainders. This will be the constructive and explicit part of the analysis, relying on the model minimization problems. Note that this is the only part of our analysis where the concentration-compactness method will be used, to analyze model problems with vanishing magnetic fields. Then we will establish lower bounds and localization estimates. For that purpose, inspired by the linear spectral techniques (see for instance [8, Part I]), we will provide an alternative point of view to the semiclassical concentration-compactness arguments of [START_REF] Di Cosmo | Semiclassical stationary states for nonlinear Schrödinger equations with a strong external magnetic field[END_REF] by using semiclassical partitions of unity adapted simultaneously to the L p -norm and the magnetic quadratic form. Moreover, in the case of non-vanishing magnetic fields, we will establish exponential decay estimates of all the minimizers of (1.1) away from the magnetic wells (the minima of B). In fact, we will use the philosophy of the semiclassical linear methods: the more accurate the estimate of λ(Ω, A, p, h) obtained, the more refined is the localization of the bound states. A very rough localization estimate of the bound states in L p -norm (directly related to the remainders in the estimates of λ(Ω, A, p, h)) will then be enough to get an a priori control of the nonlinearity and the investigation will be reduced to the well-known semiclassical concentration estimates à la Agmon (see [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF][START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]), jointly with standard elliptic estimates. Finally, note that our investigation deals with the magnetic analog of the pure electric case of [START_REF] Wang | On concentration of positive bound states of nonlinear Schrödinger equations[END_REF] (see also [START_REF] Del Pino | Semi-classical states for nonlinear Schrödinger equations[END_REF]). We could include in our analysis an electric potential, but we refrain to do so to highlight the pure magnetic effects.

1.2. Results. We would like to provide an accurate description of the behavior of λ(Ω, A, p, h) when h goes to zero. Locally, we can approximate by either a constant magnetic field, or a magnetic field having a zero of a certain order. Therefore, we introduce the following notation.

Definition 1.3. For k ∈ N, we define (1.3) λ [k] (p) = λ(R 2 , A [k] , p, 1) = inf ψ∈Dom(Q A [k] ),ψ =0 Q A [k] (ψ) ψ 2 L p , where A [k] (x, y) = 0, x k+1 k+1 .
Here

Q A [k] (ψ) = R 2 |(-i∇ + A [k] )ψ| 2 dx, with domain
Dom(Q A [k] ) = ψ ∈ L 2 (R 2 ) : (-i∇ + A [k] )ψ ∈ L 2 (R 2 ) .
In the case k = 0 and p ≥ 2, it is known that the infimum is a minimum (see [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF]). We will prove in this paper that, for k ≥ 1 and p > 2, the minimum is also attained, even if the corresponding magnetic field does not satisfy the assumptions of [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF].

We can now state our first theorem concerning the case when the magnetic field does not vanish.

Theorem 1.4. Let p ≥ 2. Let us assume that A is smooth on Ω, that B = ∇ × A does not vanish on Ω and that its minimum b 0 is attained in Ω. Then there exist C > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ),

(1 -Ch 1 8 )λ [0] (p)b 2 p 0 h 2 h -2 p ≤ λ(Ω, A, p, h) ≤ (1 + Ch 1/2 )λ [0] (p)b 2 p 0 h 2 h -2 p .
Moreover, if the magnetic field is only assumed to be smooth and positive on Ω (with a minimum possibly on the boundary), the lower bound is still valid.

Remark 1.5. The error estimate in the upper bound in Theorem 1.4 matches the corresponding bound in the well-known linear case and we expect it to be optimal. However, the relative error of h 1 8 in the lower bound is unlikely to be best possible. The same remark applies to the error bounds in Theorem 1.8.

In the following theorem, we state an exponential concentration property of the minimizers.

Theorem 1.6. Let p > 2, ρ ∈ 0, 1 2 , ε > 0 and consider the same assumptions as in Theorem 1.4 and also assume that the minimum is unique and attained at x 0 ∈ Ω.

Then there exist C > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ) and all ψ solution of (1.2), we have

ψ L ∞ ( D(x 0 ,2ε)) ≤ Ce -Ch -ρ ψ L ∞ (Ω) ,
where D(x, R) denotes the open ball of center x and radius R > 0.

Remark 1.7. In Theorem 1.6, if the minimum of the magnetic field is non-degenerate, we can replace ε by h γ with γ > 0 sufficiently small. In Theorem 1.6, we have the same kind of results in the case of multiple wells. Theorems 1.4 and 1.6 are quantitative improvements of [5, Theorem 1.1] in the pure magnetic case. We can notice that, when p > 2, we have

(1.4) (-ih∇ + A) 2 ϕ = |ϕ| p-2 ϕ, with ϕ = λ(Ω, A, p, h) 1
p-2 ψ. Thus, we have constructed solutions of (1.4) which decay exponentially away from the magnetic wells in the semiclassical limit.

The following theorem analyzes the case when the magnetic field vanishes along a smooth curve.

Theorem 1.8. Let p > 2. Let us assume that A is smooth on Ω, that

Γ = {x ∈ Ω : B(x) = 0},
satisfies that Γ ⊂ Ω is a smooth, simple and closed curve, and that B vanishes non-degenerately along Γ in the sense that

∇B(x) = 0, for all x ∈ Γ.
Assuming that B is positive inside Γ and negative outside, we denote by γ 0 > 0 the minimum of the normal derivative of B with respect to Γ. Then there exist C > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ),

(1 -Ch 1 33
)λ [1] (p)γ

4 3p 0 h 2 h -4 3p ≤ λ(Ω, A, p, h) ≤ (1 + Ch 1 3
)λ [1] (p)γ

4 3p 0 h 2 h -4 3p .
Remark 1.9. The case p = 2 is treated in [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF] (see also [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF][START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF]). In [START_REF] Di Cosmo | Semiclassical stationary states for nonlinear Schrödinger equations with a strong external magnetic field[END_REF], it is only stated that h -2+ 2 p λ(Ω, A, p, h) goes to zero when h goes to zero. Moreover, by using the strategy of the proof of Theorem 1.6, one can establish an exponential decay of the ground states away from Γ. 1.3. Organization of the paper. In Section 2, we investigate the existence of minimizers of (1.3) and their decay properties. In Section 3, we provide the upper bounds stated in Theorems 1.4 and 1.8. Section 4 is devoted to the analysis of the corresponding lower bounds and to the proof of Theorem 1.6.

Minimizers of the models and exponential decay

2.1. Existence of bound states and exponential decay. We first recall the diamagnetic inequality and the so-called "IMS" formula (see [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF][START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]).

Lemma 2.1. We have, for u ∈ Dom(Q A ),

|∇|u|| ≤ |(-i∇ + A)u|, a.e. which implies that ∇|u| 2 L 2 (R 2 ) ≤ Q A (u), Lemma 2.2.
If χ is a Lipschitzian function and u ∈ Dom(L h,A ), then we have

Re L h,A u, χ 2 u L 2 (Ω) = Q h,A (χu) -h 2 ∇χ u 2 L 2 (Ω) (2.1)
Finally, we recall the following useful lower bound (see for instance [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF]),

Q h,A (u) ≥ h Ω B|u| 2 dx (2.2)
for all u ∈ H 1 0 (Ω). We recall that we can define the Friedrichs extension of the electro-magnetic Laplacian as soon as the electric potential belongs to some L q space. Proposition 2.3. Let us consider A ∈ C ∞ (R 2 ) and V ∈ L q (R 2 ), for some q > 1. For all u ∈ Dom(Q A ), we may define

Q A,V (u) = R 2 |(-i∇ + A)u| 2 dx + R 2 V |u| 2 dx.
Then, for all ε > 0, there exists C > 0 such that

(2.3) ∀u ∈ Dom(Q A ), Q A,V (u) ≥ (1 -ε)Q A (u) -C u 2 L 2 (R 2 ) , Furthermore, for all ε > 0 there exists R > 0 such that ∀u ∈ Dom(Q A ) with supp u ⊂ D(0, R), (2.4)
Q A,V (u) ≥ (1 -ε)Q A (u) -ε u 2 L 2 (R 2 )
. Moreover, we may define the self-adjoint operator-the Friedrichs extension-L A,V of Q A,V whose domain is

Dom(L A,V ) = u ∈ Dom(Q A ) : (-i∇ + A) 2 + V u ∈ L 2 (R 2 ) , and L A,V u = (-i∇ + A) 2 + V u, for all u ∈ Dom(L A,V ). Proof. Let us recall the Sobolev embedding H 1 (R 2 ) ⊂ L r (R 2 ): For all r ≥ 2 there exist C(r) > 0 such that for all u ∈ H 1 (R 2 ), u L r (R 2 ) ≤ C(r)( u L 2 (R 2 ) + ∇u L 2 (R 2 ) ). (2.5)
In particular, for all v ∈ H 1 (R 2 ) and all ε > 0, we apply this inequality to the rescaled function

u ε (x) = v(ε r 2 x) to infer the rescaled version of the Sobolev embedding (2.6) v L r (R 2 ) ≤ C(r)(ε 1-r 2 v L 2 (R 2 ) + ε ∇v L 2 (R 2 )
). With the diamagnetic inequality, this implies that, for all u ∈ Dom(Q A ), we have u ∈ L r (R 2 ) for all r ≥ 2 and Q A,V (u) is well defined. Then let us prove (2.3). We use the Cauchy-Schwarz inequality to get, for all u ∈ Dom(Q A ),

R 2 V |u| 2 dx ≤ V L q (R 2 u 2 L q (R 2 ) = V L q (R 2 u 2 L 2q (R 2 ) ,
where 1 q + 1 q = 1. Since q > 1, we have 1 < q < +∞ so that with (2.6),

u 2 L 2q (R 2 ) ≤ C(q )(ε 1-q u 2 L 2 (R 2 ) + ε ∇|u| 2 L 2 (R 2 ) ) and so u 2 L 2q (R 2 ) ≤ C(q )(ε 1-q u 2 L 2 (R 2 ) + εQ A (u) + ε u 2 L 2 (R 2 ) )
and (2.3) follows as well as the existence of the Friedrichs extension L A,V .

Let us now prove (2.4). For all u ∈ Dom(Q A ) supported in D(0, R),

R 2 V |u| 2 dx ≤ V L q ( D(0,R)) u 2 L 2q (R 2 ) ,
and we deduce

R 2 V |u| 2 dx ≤ C 2 V L q ( D(0,R) |u| 2 H 1 (R 2 ) = C 2 V L q ( D(0,R)) u 2 L 2 (R 2 ) + R 2 |(-i∇ + A)u| 2 dx .
It remains to use that V ∈ L q (R 2 ) and to take R large enough to get (2.4).

Proposition 2.4. For k = 0 and p ≥ 2, the infimum in

(1.3) is a minimum. Moreover, if ψ is a minimizer, there exist η, C > 0 such that e 2η|x| (|ψ(x)| 2 + |(-i∇ + A(x))ψ)(x)| 2 ) dx ≤ C ψ 2 L 2 (R 2 ) . (2.7)
Proof. The fact that the infimum is attained is proved in [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF]. Let ψ be a minimizer such that ψ L p (R 2 ) = 1.

We introduce the potential V = -λ 0 |ψ| p-2 ≤ 0 which-by (2.5)-belongs to L q (R 2 ) for all q ≥ 2 p-2 . By Proposition 2.3, we may consider the electro-magnetic Laplacian L A [0] ,V defined as the Friedrichs extension of the quadratic form

Q A [0] ,V (u) = R 2 |(-i∇ + A [0] )u| 2 dx + R 2 V |u| 2 dx, ∀u ∈ C ∞ 0 (R 2 ).
We notice that ψ ∈ Dom(L A [0] ,V ), ψ = 0 and L A 0 ,V ψ = 0. With (2.4) in Proposition 2.3, for all ε > 0, there exists R 0 > 0 such that, for all u ∈ Dom(Q

A [0]
), such that supp (u) ⊂ D(0, R 0 ), we have

Q A [0] ,V (u) ≥ (1 -ε)Q A [0] (u) -ε u 2 L 2 (R 2 ) . But we have Q A [0] (u) ≥ u 2 L 2 (R 2 ) , so that Q A [0] ,V (u) ≥ (1 -2ε) u 2 L 2 (R 2 )
. From Persson's theorem, we infer that inf sp ess (L A [0] ,V ) ≥ 1-with sp ess denoting the essential spectrum. Now, by definition, ψ is an eigenfunction associated with the eigenvalue 0 < 1 and, by Agmon estimates (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF][START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF]), it has an exponential decay.

Corollary 2.5. Let k ∈ N and let ψ be a minimizer of (1.3).

• For any q ∈ [2, ∞), we have ψ L 2 (R 2 ) ≤ C q ψ L q (R 2 ) ≤ C q ψ L 2 (R 2 ) . • For any q ∈ [2, ∞), we have e η|x| ψ ∈ L q (R 2 ) and e η|x| ψ L q (R 2 ) ≤ C ψ L q (R 2 ) .
Proof. We give the proof for k = 0. The proof for k ≥ 1 is identical using Proposition 2.7 below instead of Proposition 2.4.

It follows from Proposition 2.4 and the diamagnetic inequality (Lemma 2.1) that a minimizer

ψ satisfies |ψ| H 1 (R 2 ) ≤ C ψ L 2 (R 2 )
. Therefore, we get by the Sobolev inequality (2.5) that

ψ L q (R 2 ) ≤ C ψ L 2 (R 2 ) ,
for any q ∈ [2, ∞). On the other hand, we can use the Hölder inequality, followed by the previous inequality, to estimate

ψ 2 L 2 (R 2 ) ≤ ψ L q (R 2 ) ψ L q (R 2 ) ≤ C ψ L q (R 2 ) ψ L 2 (R 2 )
. This proves the first parts of the corollary.

The exponential bound in L q now follows from Proposition 2.4 and the diamagnetic inequality (Lemma 2.1) and the Sobolev inequality (2.5).

We will need the following lemma.

Lemma 2.6. Let us consider R > 0 and a family of smooth vector potentials

(A n ) n≥0 on D(0, R) such that B n = ∇ × A n → +∞ uniformly on D(0, R).Then, the lowest eigenvalue λ Neu 1 (D(0, R), A n ) of the Neumann realization of the magnetic Laplacian (-i∇ + A n ) 2 on D(0, R) tends to +∞.
Proof. We start by introducing an auxiliary operator. Let R/2 ≤ r ≤ R and consider the non-magnetic Laplace operator -∆ on the annulus D(0, R) \ D(0, r) with Dirichlet condition at r and Neumann condition at R. If we let ζ(r) be the lowest eigenvalue of this operator, then it is a simple fact that (R -r) 2 ζ(r) ≥ δ 0 > 0 for some δ 0 independent of r.

Let ψ n be an L 2 -normalized ground state of the Neumann realization of the magnetic Laplacian (-i∇ + A n ) 2 on D(0, R). We let q n denote the quadratic form of (-i∇ + A n ) 2 .

Assume for contradiction that λ Neu 1 (D(0, R), A n ) remains bounded (along a subsequence). Let r < R and define m n (r) = D(0,r) |ψ n (x)| 2 dx. We start by proving that m n (r) → 0 as n → ∞. In order to prove this, let us consider a partition of unity with χ 2 1 + χ 2 2 = 1, supp χ 1 ⊂ D(0, r + R-r 2 ), χ 1 = 1 on D(0, r) and such that

|∇χ 1 | 2 + |∇χ 2 | 2 ≤ C (R-r) 2 .
By the IMS-formula (2.1) we have

q n (ψ n ) ≥ q n (χ 1 ψ n ) + q n (χ 2 ψ n ) - C |R -r| 2 {r≤|x|≤r+ R-r 2 } |ψ n | 2 dx. (2.8)
Since χ 1 ψ n has compact support in D(0, R), we can estimate, using (2.2),

q n (χ 1 ψ n ) ≥ B n |χ 1 ψ n | 2 dx ≥ (inf B n )m n (r). (2.9)
Since, by assumption, q n (ψ n ) is bounded, we conclude that m n (r) → 0.

But now we can reconsider (2.8) to get, with the diamagnetic inequality,

q n (ψ n ) ≥ ζ(r) χ 2 ψ n 2 2 - Cm n (r + R-r 2 ) |R -r| 2 . (2.10) Since m(r + R-r 2 ) → 0 and χ 2 ψ n 2 2 → 1 as n → ∞, we find lim inf n→∞ q n (ψ n ) ≥ ζ(r). (2.11) But ζ(r) → ∞ as r → R, which is a contradiction. Proposition 2.7. For k ≥ 1 and p > 2, the infimum in (1.3) is a minimum. Moreover, if ψ is a minimizer, there exist η, C > 0 such that e 2η|x| (|ψ(x)| 2 + |(-i∇ + A(x))ψ)(x)| 2 ) dx ≤ C ψ 2 L 2 (R 2 ) . (2.12)
Proof. The existence of a minimizer is not a consequence of the results in [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF]. Nevertheless we will also use the concentration-compactness method. For simplicity of notation we will write A instead of A [k] in this proof. Since k ≥ 1 is fixed there is no room for confusion.

Let us consider a minimizing sequence (u n ), with u n L p (R 2 ) = 1. We introduce the density measure

µ n = (|u n | 2 + |(-i∇ + A)u n | 2 ) dx whose total mass µ n (R 2
) is bounded and we can assume that it converges to some µ > 0 up to the extraction of a subsequence. Indeed, if µ = 0, by the diamagnetic inequality, we would get that (|u n |) goes to 0 in H 1 (R 2 ) and thus in L p (R 2 ).

Since p > 2, as in [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF] and using [16, Lemma 1], we are easily reduced to the "tightness" case (see Appendix A). In other words we may find a sequence

x n = (x n , y n ) such that (2.13) ∀ε > 0, ∃R > 0, ∀n ≥ 1, µ n ( D(x n , R)) ≤ ε.
We introduce the translated function ûn

(x) = u n (x + x n ) and A n (x) = A(x + x n ).
Notice at this point that with our choice of vector potential A only depends on x = (x, y) through the first coordinate x. We have

Q A (u n ) = Q An (û n ) = R 2 |D x ûn | 2 + D y + 1 k + 1 (x -x n ) k+1 ûn 2 dx.
From (2.13), we get D(0,R) |û n | 2 dx ≤ ε, and we also have

D(0,R) |D x ûn | 2 + D y + 1 k + 1 (x -x n ) k+1 ûn 2 dx ≤ C,
for some C independent of n. By the min-max principle, we have

λ Neu 1 (D(0, R), A n ) D(0,R) |û n | 2 dx ≤ D(0,R) |D x ûn | 2 + D y + 1 k + 1 (x -x n ) k+1 ûn 2 dx.
If |x n | → +∞, we get, by Lemma 2.6 that λ Neu 1 (D(0, R), A n ) → +∞ and thus there exists N ≥ 1 such that, for all n ≥ N ,

D(0,R) |û n | 2 dx ≤ ε.
We infer that (u n ) tends to 0 in L 2 (R 2 ). Since the diamagnetic inequality implies that (|u n |) n≥0 is bounded in H 1 , we get a contradiction to the assumption that u n p = 1 by using a Sobolev embedding. Therefore we may assume that x n converges to some x * ∈ R. From this we infer that (∇û n ) is bounded in H 1 (D(0, R)) and since D(0,R) |û n | 2 dx ≤ ε, we can use the Rellich's Criterion [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]Theorem XIII.65] to see that we may assume that ûn converges in L 2 (R 2 ) to some û * . In fact, the relative compactness is also verified in L q (R 2 ) with q ≥ 2, since the Hölder inequality provides

D(0,R) |û n | q dx ≤ D(0,R) |û n | 2 dx 1 2 D(0,R) |û n | 2(q-1) dx 1 2
and that, by diamagnetism,

D(0,R) |û n | 2(q-1) dx ≤ ûn q-1 L 2q-2 (R 2 ) ≤ CQ An (û n ) q-1 ≤ C.
Therefore we may assume that ûn converges in L p (R 2 ) so that we deduce, by trans-

lation invariance, 1 = u n L p (R 2 ) = ûn L p (R 2 ) → û * L p (R 2 ) and thus û * L p (R 2 ) = 1.
Moreover, up to extractions of subsequences and a diagonal argument, we can assume that (û n ) converges to û * weakly in H 1 loc (R 2 ). Then, we can conclude the proof. Indeed, we have, for all R > 0 and n ≥ 1,

Q A (u n ) = Q An (û n ) ≥ D(0,R) |D x ûn | 2 + D y + 1 k + 1 (x -x n ) k+1 ûn 2 dx
so that, due to the weak convergence in

H 1 loc (R 2 ), lim inf n→+∞ Q A (u n ) ≥ D(0,R) |D x û * | 2 + D y + 1 k + 1 (x -x * ) k+1 û * 2 dx.
Therefore, lim inf

n→+∞ Q A (u n ) ≥ Q A * (û * ) = Q A (u * ), where u * (x, y) = û * (x + x * , y), A * (x, y) = (0, 1 k+1 (x -x * ) k+1
), and we also have

u * L p (R 2 ) = 1.
To stress the difference between the linear (p = 2) and the non-linear problem (p > 2) we include the following simple result. Proposition 2.8. For k ≥ 1 and p = 2, the infimum in (1.3) is not a minimum.

Proof. For α ∈ R and k ≥ 1, we define the 'Montgomery operator' (or anharmonic oscillator) of order k,

H (k) (α) = D 2 t + t k+1 k + 1 -α 2 ,
as a self-adjoint operator in L 2 (R). Let λ 1,H (k) (α) be its ground state eigenvalue. By [9, Theorem 1.3], for all k ≥ 1, there exists a unique point α

[k] 0 ∈ R such that the function α → λ 1,H (k) (α) attains its minimum at α [k] 0 . Also λ 1,H (k) (α) → ∞ as α → ∞.
By partial Fourier transform in the y-coordinate and thanks to [21, Theorem XIII.85], we get

λ [k] (p = 2) = λ 1,H (k) (α [k] 0 ) . Suppose now by contradiction that ψ is an L 2 -normalized eigenfunction of the mag- netic Laplacian (-i∇ + A [k] ) 2 corresponding to λ [k] (p = 2). Let ψ(x, α) ∈ L 2 (R 2
) be the partial Fourier transform of ψ in the y variable. In particular,

λ [k] (2) = R R |D x ψ| 2 + α - x k+1 k + 1 ψ 2 dx dα. (2.14)
By normalization and choosing δ > 0 small enough, we may assume that we have

{|α-α [k] 0 |≥δ} | ψ(x, α)| 2 dx dα ≥ 1/2.
Using the continuity with respect to α and the uniqueness of the minimum, there exists ε > 0 such that inf {|α-α

[k] 0 |≥δ} λ 1,H (k) (α) ≥ λ 1,H (k) (α [k] 0 ) + ε,
and so we get

λ [k] (2) ≥ R 2 λ 1,H (k) (α) | ψ(x, α)| 2 dx dα ≥ (λ 1,H (k) (α [k] 0 ) +ε) {|α-α [k] 0 |≥δ} | ψ(x, α)| 2 dx dα+λ 1,H (k) (α [k] 0 ) {|α-α [k] 0 |<δ} | ψ(x, α)| 2 dx dα
and thus

λ [k] (2) ≥ λ 1,H (k) (α [k] 0 ) + ε 2 .
This is a contradiction and finishes the proof.

Upper bounds

This section is devoted to the proof of the upper bounds in Theorems 1.4 and 1.8.

3.1. Non-vanishing magnetic field. In this section we work under the assumptions of Theorem 1.4. Let us consider v a minimizer associated with (1.3) for k = 0 and let

ψ(x) = h -1 p e i φ(x) h χ(x)v x -x 0 h 1 2
.

Here x 0 denotes a point in Ω where the minimum of the magnetic field is obtained, χ ∈ C ∞ 0 (Ω), with χ ≡ 1 in a neighborhood of x 0 , and φ is a real function such that à = A + ∇φ satisfies in a fixed neighborhood of x 0 :

Ã(x) -b 0 Ã[0] (x) ≤ C|x -x 0 | 2 , Ã[0] (x) = A [0] (x -x 0 )
We have, by Corollary 2.5,

Ω |ψ(x)| p dx = R 2 χ p x 0 + h 1 2 y |v(y)| p dy = R 2 |v(y)| p dy + O(h ∞ ) v p L p (R 2 )
and, with the "IMS" formula,

Q h,A (ψ) = h -2 p Ω χ 2 x 0 + h 1 2 y -ih∇ + Ã v x -x 0 h 1 2 2 dx + O(h ∞ ) v 2 L 2 (R 2 )
so that, for all ε > 0,

h 2 p Q h,A (ψ) ≤ (1 + ε) Ω -ih∇ + b 0 Ã[0] v x -x 0 h 1 2 2 dx + (1 + ε -1 ) Ω Ã -b 0 Ã[0] v x -x 0 h 1 2 2 dx + O(h ∞ ) v 2 L 2 (R 2 ) .
Due to the exponential decay of v given in Proposition 2.4, we have

R 2 |y| 4 |v(y)| 2 dy < +∞,
and thus

h 2 p Q h,A (ψ) ≤ (1 + ε)h 2 R 2 -i∇ + b 0 Ã[0] v(y) 2 dy + C 2 (1 + ε -1 )h 3 R 2 |v(y)| 2 dx + O(h ∞ ) v 2 L 2 (R 2 ) .
We have by (2.2),

R 2 -i∇ + b 0 Ã[0] v(y) 2 dy ≥ b 0 R 2 |v(y)| 2 dy.
We deduce the upper bound:

h 2 p Q h,A (ψ) ≤ (1 + ε)h 2 + b -1 0 C 2 (1 + ε -1 )h 3 R 2 -i∇ + b 0 Ã[0] v(y) 2 dy.
We take ε = h 1/2 so that,

h 2 p λ(Ω, A, p, h) ≤ h 2 + Ch 5/2 R 2 -i∇ + b 0 Ã[0] v(y) 2 dy R 2 |v(y)| p dy 2 p
.

We get

λ(Ω, A, p, h) ≤ h -2 p h 2 + Ch 5/2 λ(1, b 0 Ã[0] , p).
By homogeneity and gauge invariance, we have

λ(R 2 , b 0 Ã[0] , p, 1) = b 2 p 0 λ(R 2 , A [0]
, p, 1). We infer the upper bound

λ(Ω, A, p, h) ≤ h -2 p b 2 p 0 h 2 λ(R 2 , A [0] , p, 1) + Ch 5 2 , (3.1)
So the upper bound of Theorem 1.4 is proved.

Vanishing magnetic field.

Let us now work under the assumptions of Theorem 1.8. We can define the standard tubular coordinates in a neighborhood of a point x 0 ∈ Γ which minimizes the normal derivative of B, Γ x → ∂ n,Γ B(x). These coordinates are defined through the local diffeomorphism Φ : (s, t) → c(s)+tn(c(s)) = x where c is a parametrization of Γ such that |c (s)| = 1 and n(c(s)) is the inward pointing normal of Γ at c(s), that is det(c (s), n(c(s))) = 1. We may assume that Φ(0, 0) = x 0 . For further details, we refer to [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Appendix F]. In these new coordinates the quadratic form becomes, for functions ψ supported near x 0 ,

Q h,A (ψ) = Q h,A ( ψ) = |hD t ψ| 2 + (1 -tk(s)) -2 |(hD s + Ã) ψ| 2 (1 -tk(s)) ds dt, (3.2)
with ψ(s, t) = e iϕ(s,t)/h ψ(Φ(s, t)), where ϕ corresponds to a local change of gauge. Moreover we have let k(s) = γ (s) • n(γ(s)), and

Ã(s, t) = - t 0 (1 -uk(s)) B(s, u) du, with B(s, t) = B(Φ(s, t)).
Note that we also have, as soon as ψ is supported near Γ,

Ω |ψ| p dx = | ψ| p (1 -tk(s)) ds dt.
We may write the following Taylor estimate

Ã(s, t) + γ 0 t 2 2 ≤ C(|s| 3 + |t| 3 ).
Let us consider w the complex conjugate of a minimizer associated with (1.3) for k = 1, normalized in L p (R 2 ) and let

ψ(s, t) = h -2 3p γ 2 3p 0 χ(s, t)w γ 1 3 0 s h 1 3
, γ

1 3 0 t h 1 3
, where χ ∈ C ∞ 0 (R 2 ), χ ≡ 1 near 0, with supp χ sufficiently small. We have, using the exponential decay of w,

| ψ(s, t)| p (1-tk(s)) ds dt = |w(σ, τ )| p (1-τ γ -1 3 0 h 1 3 k(h 1 3 γ -1 3 0 σ)) dσ dτ +O(h ∞ ) w p L p (R 2 ) so that, | ψ(s, t)| p (1 -tk(s)) ds dt ≥ (1 -Ch 1 3 ) |w(σ, τ )| p dσ dτ.
Thanks to support considerations, we get With the exponential decay of w, we have

|hD t ψ| 2 + (1 -tk(s)) -2 |(hD s + Ã) ψ| 2 (1 -tk(s)) ds dt
|t| |hD t ψ| 2 + |(hD s + Ã) ψ| 2 ds dt ≤ Ch 5 3 h -4 3p h 2 3 .
In the same way we get

|hD t ψ| 2 + |(hD s + Ã) ψ| 2 ds dt ≤ |hD t ψ| 2 + hD s -γ 0 t 2 2 ψ 2 ds dt + Ch 5 3 h -4 3p h 2 3
and

|hD t ψ| 2 + hD s -γ 0 t 2 2 ψ 2 ds dt = γ 4 3p 0 h 4 3 h 2 3 h -4 3p λ [1] (p) + O(h ∞ ).
We deduce

Q h,A (ψ) ψ 2 L p (R 2 ) ≤ (1 + Ch 1 3 )γ 4 3p 0 h 2 h -4 3p λ [1] (p)
and the conclusion immediately follows.

Lower bounds

We are now interested in the lower bounds in Theorems 1.4 and 1.8.

4.1.

Quadratic partition of unity and reconstruction of L p -norm. Let us introduce a quadratic partition of unity "with small interaction supports". In the following we will for notational convenience use the

∞-norm on R 2 , explicitly |x| ∞ = max(|x 1 |, |x 2 |). Lemma 4.1. Let us consider E = {(α, ρ, h, ) ∈ (R + ) 3 × Z 2 : α ≥ ρ}.
There exists a family of smooth cutoff functions (χ

[ ] α,ρ,h ) (α,ρ,h, )∈E on R 2 such that 0 ≤ χ [ ] α,ρ,h ≤ 1, χ [ ] α,ρ,h = 1, on |x -(2h ρ + h α ) | ∞ ≤ h ρ , χ α,ρ,h = 0, on |x -h ρ | ∞ ≥ h ρ + h α ,
and such that

∈Z 2 χ [ ] α,ρ,h 2 = 1.
Moreover there exists D > 0 such that, for all h > 0, (4.1)

∈Z 2 |∇χ [ ] α,ρ,h | 2 ≤ Dh -2α . Proof. Let us consider F = {(α, ρ, h) ∈ (R + ) 3 : α ≥ ρ}.
There exists a family of smooth cutoff functions of one real variable (χ α,ρ,h

) (α,ρ,h)∈F such that 0 ≤ χ α,ρ,h ≤ 1, χ α,ρ,h = 1 on |x| ≤ h ρ + 1 2 h α and χ α,ρ,h = 0 on |x| ≥ h ρ + h α ,
and such that for all (α, ρ) with α ≥ ρ > 0, there exists C > 0 such that for all h > 0, |∇χ α,ρ,h | ≤ Ch -α . Then, we define :

S α,ρ,h (x) = ∈Z 2 χ 2 α,ρ,h x 1 -(2h ρ + h α ) 1 χ 2 α,ρ,h x 2 -(2h ρ + h α ) 2 ,
and we have

∀x ∈ R 2 , 1 ≤ S α,ρ,h (x) ≤ 4.
We let

χ [ ] α,ρ,h (x) = χ α,ρ,h (x 1 -(2h ρ + h α ) 1 )χ α,ρ,h (x 2 -(2h ρ + h α ) 2 ) S α,ρ,h (x) ,
which satisfies the wished estimates by standard arguments.

Given a grid and a non-negative and integrable function f , the following lemma states that, up to a translation of the grid, the mass of f carried by a slightly thickened grid is controlled by a slight fraction of the total mass of f . Lemma 4.2. For r > 0 and δ > 0, we define the grid Λ r = ((rZ) × R) ∪ (R × (rZ)) and the thickened grid

Λ r,δ = {x ∈ R 2 : dist(x, Λ r ) ≤ δ}.
Let us consider a non-negative function f belonging to L 1 (R 2 ). Then there exists τ (r, δ, f ) = τ ∈ R 2 such that :

Λ r,δ +τ f (x) dx ≤ 3δ r + 2δ R 2 f (x) dx.
Proof. We let e = 1 √ 2 (1, 1). We notice that

r 2δ +1 j=0 Λ r,δ +jδe f (x) dx = R 2 g r,δ (x)f (x) dx, with g r,δ (x) = r 2δ +1 j=0 1 Λ δ +jδe (x).
We have, for almost all x, g r,δ (x) ≤ 3, so that we get

r 2δ +1 j=0 Λ r,δ +jδe f (x) dx ≤ 3 R 2 f (x) dx.
Therefore, there exists j ∈ 0, . . . , r δ + 1 , such that

Λ r,δ +jδe f (x) dx ≤ 3 r 2δ + 2 R 2 f (x)
and the conclusion easily follows.

We can now establish the following lemma which permits to recover the total L p -norm from the local contributions defined by the quadratic partition of unity. α,ρ,h ) defined in Lemma 4.1, with α > ρ > 0. There exist C > 0 and h 0 > 0 such that for all ψ ∈ L p (Ω) and h ∈ (0, h 0 ), there exists

τ α,ρ,h,ψ = τ ∈ R 2 such that Ω | χ[ ] α,ρ,h ψ(x)| p dx ≤ Ω |ψ(x)| p dx ≤ (1 + Ch α-ρ ) Ω | χ[ ] α,ρ,h ψ(x)| p dx, with χ[ ] α,ρ,h (x) = χ[ ] α,ρ,h (x-τ ).
Moreover, the translated partition ( χ[ ] α,ρ,h ) still satisfies (4.1).

Proof. The first inequality is obvious since the cutoff functions are bounded by 1 and their squares sum to unity. For the second inequality, we write, for any translation τ ,

Ω |ψ(x)| p dx = Ω χ[ ] α,ρ, p |ψ(x)| p dx + Ω ϕ α,ρ (x)|ψ(x)| p dx, where ϕ α,ρ = χ[ ] α,ρ, 2 - χ[ ] α,ρ, p . The smooth function ϕ α,ρ is supported on τ + Λ h ρ + 1 2 h α ,2h α and Ω ϕ α,ρ (x)|ψ(x)| p dx ≤ τ +Λ h ρ + 1 2 h α ,2h α f (x) dx,
where f (x) = |ψ(x)| p for x ∈ Ω and f (x) = 0 elsewhere. Thus, by Lemma 4.2, we find τ such that

Ω ϕ α,ρ (x)|ψ(x)| p dx ≤ Ch α-ρ R 2 |ψ(x)| p dx
and the conclusion easily follows.

4.2.

Lower bound: non-vanishing magnetic field. This section is devoted to the proof of the lower bound in Theorem 1.4.

4.2.1.

A lower bound for the eigenvalue. Let us consider ψ ∈ Dom(Q h,A ). With the "IMS" formula associated with the partition of unity ( χ[ ] α,ρ,h ) that is adapted to ψ (see Lemma 4.3), we infer

Q h,A (ψ) = Q h,A ( χ[ ] α,ρ,h ψ) -h 2 ∇ χ[ ] α,ρ,h ψ 2 L 2 (Ω) .
We have

(4.2) Q h,A (ψ) ≥ Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) .
By the min-max principle, we get

(4.3) λ(Ω, A, 2, h) χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≤ Q h,A ( χ[ ] α,ρ,h ψ) and we recall that (see [13, Theorem 1.1]) (4.4) λ(Ω, A, 2, h) = b 0 h + O(h 3 2 ) so that Q h,A (ψ) ≥ (1 -Dh 1-2α ) Q h,A ( χ[ ] α,ρ,h ψ).
Then, we bound the local energies from below. Thanks to support considerations (recall that α ≥ ρ), we have, modulo a local change of gauge

e iφ [ ] /h , Q h,A ( χ[ ] α,ρ,h ψ) ≥ (1 -ε)Q h,b j A [0] (e iφ [ ] /h χ[ ] α,ρ,h ψ) -Cε -1 h 4ρ χ[ ] α,ρ,h ψ 2 L 2 (Ω)
so that it follows, by using again (4.3),

Q h,A ( χ[ ] α,ρ,h ψ) ≥ (1 -ε -Cε -1 h 4ρ-1 )Q h,b j A [0] (e iφ [ ] /h χ[ ] α,ρ,h ψ). We take ε = h 2ρ-1
2 and we deduce (4.5)

Q h,A (ψ) ≥ (1 -Dh 1-2α -Ch 2ρ-1 2 ) b 2/p h 2 h -2/p λ [0] (p) χ[ ] α,ρ,h ψ 2 L p (Ω) so that Q h,A (ψ) ≥ (1 -Dh 1-2α -Ch 2ρ-1 2 )b 2/p 0 h 2 h -2/p λ [0] (p) χ[ ] α,ρ,h ψ 2 L p (Ω) Since p ≥ 2, we have χ[ ] α,ρ,h ψ 2 L p (Ω) ≥ Ω | χ[ ] α,ρ,h ψ| p dx 2 p
.

Using Lemma 4.3, we infer

(4.6) χ[ ] α,ρ,h ψ 2 L p (Ω) ≥ (1 -Ch α-ρ ) ψ 2 L p (Ω) .
Finally, we get

Q h,A (ψ) ≥ (1 -Dh 1-2α -Ch 2ρ-1 2 )(1 -Ch α-ρ )b 2/p 0 h 2 h -2/p λ [0] (p) ψ 2 L p (Ω) .
Optimizing the remainders, we choose 1 -2α = 2ρ -1 2 = α -ρ so that ρ = 5 16 and α = 7 16 and

Q h,A (ψ) ≥ (1 -Ch 1 8 )b 2/p 0 h 2 h -2/p λ [0] (p) ψ 2 L p (Ω) . (4.7)
This gives the lower bound needed for Theorem 1.4. Combined with (3.1), (4.7) yields the proof of Theorem 1.4.

4.2.2.

A direct application to the localization. The following proposition provides a rough (but quantitative) localization in L p -norm of the minimizers near the minimum of the magnetic field. Proposition 4.4. Let us assume that B |Ω admits a unique minimum attained at 0 ∈ Ω. For all ε > 0, there exist h 0 > 0, C > 0 such that for all h ∈ (0, h 0 ) and ψ minimizer,

(4.8) ψ L p ( D(0,2ε)) ≤ Ch 1 8p ψ L p (Ω) .
In the case when the minimum is non-degenerate, this can be improved to

ψ L p ( D(0,2h ρ)) ≤ Ch ( 1 8 -2ρ) 1 p ψ L p (Ω) ,
where ρ < 1 16 . Proof. We apply (4.5) to a minimizer ψ and we get, with choices of ρ and α given in the previous section:

λ(Ω, A, p, h) ψ 2 L p (Ω) ≥ (1 -Ch 1 8 ) b 2/p h 2 h -2/p λ [0] (p) χ[ ] α,ρ,h ψ 2 L p (Ω) .
With the upper bound of Theorem 1.4 and (4.6), we get

b 2/p -b 2/p 0 χ[ ] α,ρ,h ψ 2 L p (Ω) ≤ Ch 1 8 ψ 2 L p (Ω) . (4.9)
Let us introduce K 1 (h, ε) = { : x ∈ D(0, ε)}. From (4.9) and the uniqueness of the minimum, we have (for some η > 0 and all h sufficiently small),

η / ∈K 1 (h,ε) χ[ ] α,ρ,h ψ 2 L p (Ω) ≤ Ch 1 8 ψ 2 L p (Ω) , (4.10)
and (by concavity),

/ ∈K 1 (h,ε) χ[ ] α,ρ,h ψ 2 L p (Ω) ≥   / ∈K 1 (h,ε) Ω | χ[ ] α,ρ,h ψ| p dx   2 p . so that / ∈K 1 (h,ε) Ω | χ[ ] α,ρ,h ψ| p dx ≤ Ch p 16 ψ p L p (Ω) . (4.11)
We get, using Lemma 4.3 in the last step,

/ ∈K 1 (h,ε) Ω | χ[ ] α,ρ,h ψ| p dx = / ∈K 1 (h,ε) Ω | χ[ ] α,ρ,h | 2 |ψ| p dx + / ∈K 1 (h,ε) Ω (| χ[ ] α,ρ,h | p |ψ| p -| χ[ ] α,ρ,h | 2 |ψ| p ) dx ≥ / ∈K 1 (h,ε) Ω | χ[ ] α,ρ,h | 2 |ψ| p dx -Ch 1 8 ψ p L p (Ω) . (4.12)
We infer (4.8) upon inserting (4.12) in (4.11).

If the minimum of B is non-degenerate, i.e. the Hessian of B is strictly positive, (4.10) improves to

/ ∈K 1 (h,h ρ) χ[ ] α,ρ,h ψ 2 L p (Ω) ≤ Ch 1 8 -4 ρ p ψ 2 L p (Ω) ,
with ρ < ρ = 5 16 and we get the desired improvement by the same arguments. 4.2.3. Proof of Theorem 1.6. By Proposition 4.4, there exists γ > 0 such that

ψ p L p ( D(0,2ε)) ≤ Ch γ ψ p L p (Ω) . (4.13)
We assume that ψ L p (Ω) = 1. Then, we have the following estimate of the non-linear electric potential, for v ∈ Dom(Q h,A ) and supported away from the ball D(0, 2ε),

λ(Ω, A, p, h) Ω |ψ| p-2 |v| 2 dx ≤ λ(Ω, A, p, h) v 2 L p D(0,2ε) |ψ| p dx p-2 p ≤ Cλ(Ω, A, p, h)h γ p-2 p v 2 L p (Ω)
, where we used the Hölder inequality and (4.13). We now apply (2.6) to the extension by 0 of v to get

v 2 L p (R 2 ) ≤ C(ε 1-p 2 v 2 L 2 (R 2 ) + ε ∇|v| 2 L 2 (R 2 )
) and the diamagnetic inequality implies

v 2 L p (Ω) ≤ C(ε 1-p 2 v 2 L 2 (Ω) + εh -2 Q h,A (v)) so that λ(Ω, A, p, h) Ω |ψ| p-2 |v| 2 dx ≤ Ch 2-2 p h γ p-2 p (ε 1-p 2 v 2 L 2 (Ω) + εh -2 Q h,A (v)).
Let us now choose an appropriate ε. We would like to have

h 1-2 p h γ p-2 p ε 1-p 2 1, h 2-2 p h γ p-2 p εh -2 1
We choose ε = δh

2 p +γ(-1+ 2
p ) with δ > 0 arbitrarily small. We get

h 1-2 p h γ p-2 p ε 1-p 2 = δ 1-p 2 h 1-2 p +γ p-2 p +(1-p 2 ){ 2 p +γ(-1+ 2 p )} = δ 1-p 2 h γ 2 (p-2)
. We could even choose a very small power of h for δ. We infer that, for v in the domain of the magnetic Laplacian and supported in D(0, 2ε), (4.14) λ(Ω, A, p, h)

Ω |ψ| p-2 |v| 2 dx ≤ C hδ 1-p 2 h γ 2 (p-2) v 2 L 2 (Ω) + δQ h,A (v) .
Let us now establish our Agmon estimates. We recall the equation

L h,A,V h ψ = (-ih∇ + A) 2 ψ + V h ψ = 0, with V h = -λ(Ω, A, p, h)|ψ| p-2 .
We consider a smooth cutoff function χ such that χ = 0 in D(0, 2ε), 0 ≤ χ ≤ 1 and χ = 1 on D(0, 4ε). We write the IMS formula and get Q h,A,V h (e χh -ρ |x| ψ) -Ch 2-2ρ e χh -ρ |x| ψ 2 L 2 (Ω) ≤ 0. Then, we introduce a quadratic partition of unity

χ 2 1 + χ 2 2 = 1,
such that χ 2 is supported in D(0, 2ε). With the "IMS" formula, we deduce

Q h,A,V h (χ 1 e χh -ρ |x| ψ) + Q h,A,V h (χ 2 e χh -ρ |x| ψ) -Ch 2-2ρ χ 1 e χh -ρ |x| ψ 2 L 2 (Ω) -Ch 2-2ρ χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ≤ 0.
Then, with (4.14), we have

Q h,A,V h (χ 2 e χh -ρ |x| ψ) -Ch 2-2ρ χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ≥ (1 -Cδ)Q h,A (χ 2 e χh -ρ |x| ψ) -hδ 1-p 2 h γ 2 (p-2) χ 2 e χh -ρ |x| ψ 2 L 2 (Ω)
Let us recall that

Q h,A (χ 2 e χh -ρ |x| ψ) ≥ hb 0 χ 2 e χh -ρ |x| ψ 2 L 2 (Ω)
and we deduce

Q h,A,V h (χ 2 e χh -ρ |x| ψ) -Ch 2-2ρ χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ≥ (1 -Cδ)hb 0 -Ch 2-2ρ χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ≥ ηh χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ,
as soon as ρ ∈ 0, 1 2 and for h small enough. By support considerations, we have

Q h,A,V h (χ 1 e χh -ρ |x| ψ) -Ch 2-2ρ χ 1 e χh -ρ |x| ψ 2 L 2 (Ω) = Q h,A,V h (χ 1 ψ) -Ch 2-2ρ χ 1 ψ 2 L 2 (Ω)
and then

Q h,A,V h (χ 1 ψ) -Ch 2-2ρ χ 1 ψ 2 L 2 (Ω) ≥ 1 2 Q h,A (χ 1 ψ) -Ch 2-2 p Ω |ψ| p-2 |χ 1 ψ| 2 dx.
Moreover, since ψ is L p -normalized, we have

Ω |ψ| p-2 |χ 1 ψ| 2 dx ≤ C χ 1 ψ 2 L p (Ω) .
By using again the rescaled Sobolev embedding (ε = δh 2 , with δ small enough) and the diamagnetic inequality, we infer

χ 1 ψ 2 L p (Ω) ≤ C (δh 2 ) 1-p 2 χ 1 ψ 2 L 2 (Ω) + δQ h,A (χ 1 ψ) . Therefore, it follows that ηh χ 2 e χh -ρ |x| ψ 2 L 2 (Ω) ≤ C(δh 2 ) 1-p 2 ψ 2 L 2 (Ω) . and thus e χh -ρ |x| ψ 2 L 2 (Ω) ≤ Ch 1-p ψ 2 L 2 (Ω)
. With the previous analysis, we also infer that

Q h,A (χ 2 e χh -ρ |x| ψ) ≤ Ch -γ ψ 2 L 2 (Ω)
, for some γ > 0. With the same kind of arguments, we have

Q h,A (χ 1 e χh -ρ |x| ψ) ≤ CQ h,A,V h (χ 1 e χh -ρ |x| ψ) + Ch -γ ψ 2 L 2 (Ω) ≤ Ch -γ ψ 2 L 2 (Ω) . With the "IMS" formula, we find Q h,A (e χh -ρ |x| ψ) ≤ Ch -γ ψ 2 L 2 (Ω) .
Since Ω is bounded, A is regular and changing γ, we get

∇ e χh -ρ |x| ψ 2 L 2 (Ω) ≤ Ch -γ ψ 2 L 2 (Ω)
and then,

e χh -ρ |x| ψ 2 H 1 (Ω) ≤ Ch -γ ψ 2 L 2 (Ω) , e χh -ρ |x| ψ 2 L q (Ω) ≤ Ch -γ ψ 2 L 2 (Ω) ≤ Ch -γ ψ 2 L q (Ω) ,
the second inequality coming from the Sobolev embedding for all q ≥ 2 and the Hölder inequality (Ω is bounded). Finally, using the equation satisfied by ψ, we infer that

e χh -ρ |x| ψ 2 H 2 (Ω) ≤ Ch -γ ψ 2 L 2 (Ω)
and thus

e χh -ρ |x| ψ 2 L ∞ (Ω) ≤ Ch -γ ψ 2 L 2 (Ω)
. This finishes the proof of Theorem 1.6.

A b,c 1 ,c 2 (s, t) = -bt + c 1 st + c 2 2 t 2 , A b,c 1 ,c 2 = (A b,c 1 ,c 2 , 0). For p ≥ 2, we consider µ(b, c 1 , c 2 , p) = inf ψ∈H 1 A ,ψ =0 R 2 |D t ψ| 2 + |(D s + A b,c 1 ,c 2 (s, t))ψ| 2 ds dt ψ 2 L p (R 2 )
.

Then, we have (with c being the Euclidean norm of (c 1 , c 2 ))

µ(b, c 1 , c 2 , p) = µ(0, c 1 , c 2 , p) = c 4 3p
λ [1] (p).

Moreover, for p > 2, the infimum is a minimum.

Proof. It is enough to observe that

A b,c 1 ,c 2 (s, t) = c 1 st + c 2 2 t - b c 2 2 - b 2 2c 2
and to consider the translation τ = t -b c 2 . Then, a change of gauge, a rotation and a rescaling provide the conclusion.

Proof of the lower bound in Theorem 1.8. We can now prove the lower bound announced in Theorem 1.8. Let us again consider ψ ∈ Dom(Q h,A ) and use our partition of the unity (4.2). We denote by (x ) the centers of the balls and define b = B(x ). We distinguish between the balls that are close to Γ and the others by letting

J 1 (h) = { : dist(x , Γ) ≥ ε 0 }, J 2 (h) = { : h ρ < dist(x , Γ) < ε 0 }, J 3 (h) = { : dist(x , Γ) ≤ h ρ}.
Here ε 0 is chosen so small that the local coordinates (s, t) introduced around (3.2) make sense in the regions covered by J 2 (h) and J 3 (h).

4.3.1.

Collecting the balls of the region J 1 (h). Using that the magnetic field does not vanish in the region determined by J 1 (h) and using (4.4), we find first

∈J 1 (h) Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≥ ∈J 1 (h) (1-Dh 1-2α )Q h,A ( χ[ ] α,ρ,h ψ),
and then, with the lower bound of Theorem 1.4 (in the region where the magnetic field is bounded from below by a positive constant),

∈J 1 (h) Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≥ (1 -Dh 1-2α )(1 -Ch 1 8 )λ [0] (p)b 2 p 0 h 2 h -2 p χ[ ] α,ρ,h ψ 2 L p (Ω)
, where b 0 > 0 is the minimal value of the magnetic field strength in the region covered by balls from J 1 (h).

In the regions determined by J 2 (h) and J 3 (h), we use the tubular coordinates near the zero line Γ. We recall the asymptotic expansion of the linear eigenvalue (see [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF][START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF]) under the assumptions of Theorem 1.8:

(4.15) λ(Ω, A, 2, h) = γ 2 3
0 λ [1] (2)h

4 3 + o(h 4 3
).

Therefore, since we have and where R 3, is Taylor remainder of order 3 of à at (s , t ). Actually, at this point we could include the terms of order 2 in the remainder, but since we will use the higher precision in the treatment of the J 3 (h)-terms, we introduce the notation here. We deduce that

λ(Ω, A, 2, h) χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≤ Q h,A ( χ[ ] α,ρ,h ψ) we get, for j ∈ {2, 3}, (4.16) ∈J j (h) Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≥ ∈J j (h) (1 -Dh 2 3 -2α )Q h,A ( χ[ ] α,ρ,h ψ). 4
Q h,A ( χ[ ] α,ρ,h ψ) ≥ 1 2 (1 -η) -Cη -1 h 4ρ-1-ρ |hD t ψ | 2 + |(hD s -b t) ψ | 2 ds dt. Choosing η = h 2ρ-1 2 -ρ 2 , Q h,A ( χ[ ] α,ρ,h ψ) ≥ 1 2 -Ch 2ρ-1 2 -ρ 2 |hD t ψ | 2 + |(hD s -b t) ψ | 2 ds dt,
so that, by using a scaling argument and the definition of λ [1] (p),

Q h,A ( χ[ ] α,ρ,h ψ) ≥ 1 2 -Ch 2ρ-1 2 -ρ 2 b 2 p h 2-2 p λ [1] (p) ψ 2 L p ≥ 1 2 -Ch 2ρ-1 2 -ρ 2 h 2 ρ
p h 2-2 p λ [1] (p) ψ 2 L p .

We deduce that, for h sufficiently small, and using also (4.16),

∈J 2 (h) Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≥ 1 4
λ [1] (p)h [1] (2) c This gives the inequality. It follows that

∈J 3 (h) Q h,A ( χ[ ] α,ρ,h ψ) -Dh 2-2α χ[ ] α,ρ,h ψ 2 L 2 (Ω) ≥ (1 -Dh 2 3 -2α )(1 -Ch ρ)(1 -Ch 3ρ-2 3 )γ 4 3p
0 λ [1] (p)h (4.17)

0 < α < 1 3 , 0 < ρ < min 2ρ - 1 2 , 1 3 , ρ , 2 9
< ρ < α.

Under these constraints, the smallest term comes from the region determined by J 3 (h) and we find 0 λ [1] (p)h 0 λ [1] (p)h 2-

4 3p χ[ ] α,ρ,h ψ 2 L p ≥ (1 -Ch α-ρ )(1 -Dh 2 3 -2α )(1 -Ch ρ)(1 -Ch 3ρ-2 3 )γ 4 3p
0 λ [1] (p)h 2-4 3p ψ 2 L p (Ω) .

Let us consider the case when

α -ρ = 2 3 -2α = 3ρ - 2 3
which provides α = 10 33 and ρ = 8 33 . Unfortunately, the second constraint in (4.17) cannot be satisfied for this choice. Therefore, we take rather 0 λ [1] (p)h 2-4 3p ψ 2 L p (Ω) .

≤ |hD t ψ| 2 +

 2 |(hD s + Ã) ψ| 2 ds dt+C |t| |hD t ψ| 2 + |(hD s + Ã) ψ| 2 ds dt.

Lemma 4 . 3 .

 43 Let p ≥ 2. Let us consider the partition of unity (χ [ ]

4. 3 .

 3 Lower bound: vanishing magnetic field. This section is devoted to the proof of the lower bound in Theorem 1.8. Let us first state a convenient lemma. Lemma 4.5. For b, c 1 , c 2 ∈ R, with c 2 = 0, let us introduce

2 ,≥ ( 1 -≥ ( 1 -

 211 which itself comes from Lemma 4.5 with p = 2. Thus, choosing η = h 3ρ-2 3 , and using again Lemma 4.5, we getQ h,A ( χ[ ] α,ρ,h ψ) Ch ρ)(1 -Ch 3ρ-2 3 ) |hD t ( ψ )| 2 + |(hD s + A 2, (s, t))( ψ )| 2 ds dt ≥ (1 -Ch ρ)(1 -Ch 3ρ-2 3 ) c 4 3p λ [1] (p) ψ 2 L p Ch ρ)(1 -Ch 3ρ-2 3 )γwhere the last inequality will be justified below: Notice thatc 1, = ∂ s B(s, t)| (s ,t ) = ∇B(Φ(s , t )) • [c (s ) + t ∂ s n(s )] = O(h ρ), since ∇B(Φ(s , 0)) • c (s ) = 0. Similarly, c 1, = ∂ t B(s, t)| (s ,t ) = ∇B(Φ(s , t )) • n(s ) = ∂ n,Γ B(x ) + O(h ρ).

Q 2 3 -2α )( 1 -

 21 h,A (ψ) ≥ (1 -Dh Ch ρ)(1 -Ch 3ρ-

  we take ρ ∈ 0, 1 22 so thatQ h,A (ψ) ≥ (1 -Ch 1 33 )γ 4 3p

  .3.2. Collecting the balls of the region J 2 (h). ] α,ρ,h ψ)(Φ(s, t)), for some suitable local gauge transformation ϕ and withA 2, (s, t) = -b (t -t ) + c 1, (s -s )(t -t ) +

				By changing to the local coordinates
	introduced in (3.2) we find
	Q h,A (	χ[ ] α,ρ,h ψ) ≥	1 2	|hD c 2, 2	(t -t ) 2

t ψ | 2 + |(hD s + A 2, (s, t) + R 3, (s, t)) ψ | 2 ds dt,

where ψ = e iϕ /h ( χ[

  Notice that since ∈ J 2 (h), |b | ≥ Ch ρ. So we can estimate (using (2.2))|hD t ψ | 2 + |(hD s -b t) ψ | 2 ds dt ≥ hb ψ 2 L 2 ≥ Ch 1+ρ ψ 2 L 2 .

	By a Cauchy inequality and the support properties of	χ[ ] α,ρ,h ,
	|hD

t ψ | 2 + |(hD s + A 2, (s, t) + R 3, (s, t)) ψ | 2 ds dt ≥ (1 -η) |hD t ψ | 2 + |(hD s -b t) ψ | 2 ds dt -Cη -1 h 4ρ ψ 2 .

  Collecting the balls of the region J 3 (h). By changing to local coordinates as in the regionJ 2 (h), Cη -1 h 6ρ-4 3 ) |hD t ( ψ )| 2 + |(hD s + A 2, (s, t))( ψ )| 2 ds dt,where we have used|hD t ( ψ )| 2 + |(hD s + A 2, (s, t))( ψ )| 2 ds dt ≥ λ

		2 ρ p h 2-2 p	χ[ ] α,ρ,h ψ 2 L p .
		∈J 2 (h)
	4.3.3. Q h,A (	χ[ ] α,ρ,h ψ) ≥

(1 -Ch ρ) |hD t ( ψ )| 2 + |(hD s + A 2, (s, t) + R 3, (s, t))( ψ )| 2 ds dt.

We notice that (for any η > 0),

|hD t ( ψ )| 2 + |(hD s + A 2, (s, t) + R 3, (s, t))( ψ )| 2 ds dt ≥ (1 -η -

  Optimization of the parameters. The constraints on the different powers of h are:

	2-4 3p	χ[ ] α,ρ,h ψ 2 L p .
	∈J 3 (h)
	4.3.4.

3p 0 λ[START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF] (p) ψ 2 L p ,

Appendix A. Concentration-compactness method

In this section, for the convenience of the reader, we recall the strategy used in [START_REF] Esteban | Stationary solutions of nonlinear Schrödinger equations with an external magnetic field[END_REF]. We consider the minimization problem (with p > 2)

with a non-zero magnetic field B. Let us also introduce the following norm defined on

. We have λ > 0 and we introduce a minimizing sequence (u n ) n≥0 . Let us consider the density measure

By using a slight adaptation of [16, Lemma I.1], there are three possible behaviors for the sequence (µ n ) n∈N .

The first one is vanishing, that is

With the diamagnetic inequality, this implies that sup x∈R 2 |u n | H 1 (D(x,R)) → 0. By Sobolev embedding and [17, Lemma I.1], it follows that u n L p → 0. This is a contradiction.

The second possible behavior is dichotomy, that is

By using cutoff functions and the "IMS" formula, we can find χ n,1 and χ n,2 , with supports such that dist(supp (χ n,1 ), supp (χ n,2 )) → +∞ such that

and also, by Sobolev embedding,

We let α n = χ n,1 u n p L p . We may assume that (α n ) converges to some α ∈ [0, 1]. If α = 1, we get, by definition of λ, that

since (u n ) is a minimizing sequence and due to (A.2), we have Q A (χ n,2 u n ) ≤ Cε which contradicts (A.4). In the same way, we have α = 0.

With (A.2) and (A.3), we get

and thus, for α ∈ (0, 1),

p ≤ 1. This is not possible when p > 2.

The third and last possible behavior is tightness up to translation (compactness case). There exists (x n ) n≥0 such that ∀ε > 0, ∃R > 0, ∀n ≥ 1, µ n (R 2 \ D(x n , R)) ≤ ε.