Samuel Nordmann
email: samuel.nordmann@ehess.fr

Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains

Keywords: Semilinear elliptic equations, Stability, Symmetry, Neumann boundary conditions, De Giorgi's conjecture, Liouville property, Convex domains, Unbounded domains, Generalized principal eigenvalue AMS Class. No: 35B35, 35B06, 35J15, 35J61, 35B53

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland and Matano states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain Ω further satisfies Ω ∩ {|x| ≤ R} = O(R 2 ), when R → +∞. If the domain is a straight cylinder, an additional natural assumption is needed. These results can be seen as an extension to more general domains of some results on De Giorgi's conjecture.

As an application, we establish asymptotic symmetries for stable solutions when the domain satisfies a geometric property asymptotically.

Introduction

Presentation of the problem

We study some symmetry properties of stable solutions of semilinear elliptic equations with Neumann boundary conditions. We consider the following problem:

     -∆u(x) = f (u(x)) ∀x ∈ Ω, ∂ ν u(x) = 0 ∀x ∈ ∂Ω, u ∈ C 3 Ω ; ∇u ∈ L ∞ (Ω), (1)
where Ω ⊂ R n is a (possibly unbounded) smooth domain, ∂ ν is the outward normal derivative, and f is C 1 . A solution is said to be stable if the second variation of energy at u is nonnegative.

Definition 1.1 Let u be a solution of (1) and set

λ 1 := inf ψ∈C 1 0 (Ω) ψ L 2 =1 Ω |∇ψ| 2 -f (u)ψ 2 =: inf ψ∈C 1 0 (Ω) ψ L 2 =1 F(ψ), (2)
where C 1 0 (Ω) is the space of continuously differentiable functions with compact support in Ω (which do not necessarily vanish on ∂Ω).

The solution u is said to be stable if λ 1 ≥ 0, and stable non-degenerate if λ 1 > 0.

A stable non-degenerate solution is then a (non-degenerate) minimum of the energy. On the other hand, any degenerate critical point of the energy is stable according to our definition. See Appendix B for more detailed on the link between this definition and the classical dynamical definition of stability. Note that if z ∈ R is a stable root of f , i.e., f (z) = 0 and f (z) ≤ 0, it is a (trivial) stable solution. We are interested in the existence/non-existence and symmetries of non-trivial stable solutions, called patterns in the sequel. Definition 1. [START_REF] Alberti | On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property[END_REF] We call pattern (resp. non-degenerate pattern) any non-constant stable (resp. stable nondegenerate) solution.

In two independent papers, Casten, Holland [START_REF] Casten | Instability results for reaction diffusion equations with Neumann boundary conditions[END_REF], and Matano [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF] proved the following result.

Theorem 1.3 ( [20, 41])

If the domain Ω is bounded and convex, there exists no pattern to [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF].

We insist on the fact that this conclusion is valid for any f ∈ C 1 .

The main purpose of this paper is to examine whether the above theorem extends to unbounded convex domains. The classification of stable solutions in the particular case Ω = R n has already been widely investigated: this problem is intricate, and closely linked to De Giorgi's conjecture [START_REF] Wei | On De Giorgi's Conjecture: recent progress and open problems[END_REF]. However, it appears that the literature only deals, on the one hand, with bounded convex domains and, on the other hand, with the entire space R n . It does not deal, or only slightly, with general unbound convex domains.

In this paper, we establish the non-existence of non-degenerate patterns in any convex unbounded domain, and the 1-dimensional symmetry of (possibly degenerate) patterns if the domain Ω further satisfies lim sup R→+∞ |Ω∩{|x|≤R}| R 2

< +∞. In such domains, this latter result implies the non-existence of patterns if Ω is not a straight cylinder, or if f satisfies an additional natural assumption. One can see those results as extending some advances on De Giorgi's conjecture to a more general class of domains

As an application, we establish asymptotic symmetries for patterns when the domain satisfies a geometric property asymptotically. In particular, if the domain is a cylinder (with a varying section) that tends to be convex at infinity, we prove that a pattern must converge to a constant.

For the reader's convenience, we give a simple proof of Theorem 1.3 in section 2. We also recall further classical symmetry results in section 3.4, namely that stable solutions inherit from the domain's invariance with respect to translation or rotations. The appendix proposes a discussion on the notion of generalized principal eigenvalue, the different definitions of stability, and the isolation of stable solutions. In a forthcoming paper [START_REF] Nordmann | Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions[END_REF], we will investigate to what extent one can relax the assumption that the domain is convex in Theorem 1.3.

Context, general remarks, and references

Consider the energy

E ε (u) = Ω ε 2 |∇u| 2 + F (u),
where Ω is bounded, ε > 0 is a parameter and F is a two-well potential, say, F (u) := 1 2 1 -u 2 2 . Any minimizer of E ε is a stable solution of the associated Euler-Lagrange equation:

-ε 2 ∆u ε = f (u ε ) in Ω, ∂ ν u ε = 0 on ∂Ω, (3)
with Allen-Cahn's nonlinearity f (u) := u -u 3 . A series of seminal papers [START_REF] Caffarelli | Uniform convergence of a singular perturbation problem[END_REF][START_REF] Caffarelli | Phase transitions: Uniform regularity of the intermediate layers[END_REF][START_REF] Kohn | Local minimisers and singular perturbations[END_REF][START_REF] Modica | Convergence to minimal surfaces problem and global solutions of Delta u = 2(uˆ3-u)[END_REF] establishes that when ε → 0, the level sets of patterns of (3) converge to minimal surfaces in Ω. From a rescaling x ↔ εx, i.e., zooming around the origin, equation (3) at the limit ε → 0 reduces to equation [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF] in R n . According to the above, the level sets of stable solutions in R n should be minimal surfaces in R n . These are known to be necessarily hyperplanes if and only if n ≤ 7 [START_REF] Bombieri | Minimal cones and the Bernstein Problem[END_REF][START_REF] Simons | Minimal varieties in Riemannian Manifolds[END_REF]. Regarding a minimal surface which is also the graph of a function defined on R n-1 , we gain one dimension: such a surface is necessarily a hyperplane if and only if n ≤ 8 [START_REF] Bombieri | Minimal cones and the Bernstein Problem[END_REF][START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF][START_REF] Giorgi | Una estensione del teorema di Bernstein[END_REF][START_REF] Jerison | Towards a counter-example to a conjecture of De Giorgi in high dimensions[END_REF]. This brought De Giorgi to state the following conjecture.

Conjecture: (De

Giorgi) Let u be a solution of -∆u = u -u 3 in R n , such that |u| < 1 and ∂ xn u > 0.
The level sets of u are hyperplanes, at least if n ≤ 8.

The fact that the level sets of u are hyperplanes means that u is flat, i.e., u is 1-d symmetric. It should be noted that the assumption ∂ xn u > 0 implies that every level set of u is the graph of a function defined on R n-1 , which explains why the conjecture is stated for n ≤ 8 and not n ≤ 7.

A vast litterature is devoted to De Giorgi's conjecture (see [START_REF] Farina | The state of the art for a conjecture of De Giorgi and related problems[END_REF][START_REF] Wei | On De Giorgi's Conjecture: recent progress and open problems[END_REF] for a state of the art). The conjecture is proved by Ghoussoub and Gui [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF] in dimension n ≤ 2, by Ambrosio and Cabré [START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in Rˆ3 and a conjecture of De Giorgi[END_REF] in dimension n = 3, and by Savin [START_REF] Savin | Phase transitions: Regularity of flat level sets[END_REF][START_REF] Savin | Phase transitions, minimal surfaces and a conjecture of De Giorgi[END_REF] in dimension 4 ≤ n ≤ 8 under the additional assumption

lim xn→±∞ u(x , x n ) = ±1. (4)
A counter example is provided by del Pino, Kowalczyk and Wei [START_REF] Del Pino | A counterexample to a conjecture by De Giorgi in large dimensions[END_REF] in dimension n ≥ 9 (this counterexample satisfies (4)). The conjecture is still open for dimensions 4 ≤ n ≤ 8 without the additional assumption (4). Note that the proofs are, in general, valid for a rather large class of nonlinearities, and not only for Allen-Cahn's nonlinearity [START_REF] Alberti | On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property[END_REF][START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF][START_REF] Savin | Phase transitions, minimal surfaces and a conjecture of De Giorgi[END_REF]. Analogous results have also been obtained for non-compact Riemannian manifolds without boundary [START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF][START_REF] Farina | Stable Solutions of Elliptic Equations on Riemannian Manifolds[END_REF]..

The assumption ∂ xn u > 0 in De Giorgi's conjecture implies in particular that u is stable (it is a pattern). Indeed, setting v := ∂ xn u, and differentiating (1), we get -∆v -f (u)v = 0 in R n . Multiplying this equation by ψ 2 v (for a test function ψ with compact support), integrating on R n , and using the divergence theorem, we obtain

0 = R n ∇v • ∇ ψ 2 v -f (u)ψ 2 = R n 2 ψ v ∇v • ∇ψ - ψ 2 v 2 |∇v| 2 -f (u)ψ 2 ≤ R n |∇ψ| 2 -f (u)ψ 2 ,
where we use Young's inequality 2ab ≤ a 2 + b 2 in the last step. We deduce λ 1 ≥ 0, i.e., u is a pattern.

It is therefore natural to consider a variant of De Giorgi's conjecture, replacing the assumption ∂ xn u > 0 with the assumption that u is a pattern.

Conjecture: (De

Giorgi's variant) Let u be a pattern of -∆u = u -u 3 in R n , such that |u| < 1. The level sets of u are hyperplanes, at least if n ≤ 7.
Note that with the stability assumption (instead of ∂ xn u > 0), we cannot guarantee that every level set of u is the graph of a function defined on R n-1 . This explains why this conjecture is stated for n ≤ 7 (unlike De Giorgi's conjecture, stated for n ≤ 8).

This variant is proved by Ghoussoub and Gui [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF] for n ≤ 2, and a counterexample is known to exists in dimension n = 8 1 , proving that the condition n ≤ 7 is optimal. The intermediate dimensions 3 ≤ n ≤ 7 are open. Nevertheless, Dancer [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF] proves the non-existence of non-degenerate patterns in R n in any dimension.

Let us also mention the results of [START_REF] Cabré | On the stability of radial solutions of semilinear elliptic equations in all of Rˆn[END_REF] (refined in [START_REF] Villegas | Asymptotic behavior of stable radial solutions of semilinear elliptic equations in RˆN[END_REF]) which establish that radial patterns exist in R n if and only if n ≥ 11. Many further results (including regularity of weak stable solutions) are available under stronger assumptions on f , for example, if f is increasing, convex, positive, see [START_REF] Cabre | Stable solutions to semilinear elliptic equations are smooth up to dimension 9[END_REF][START_REF] Dupaigne | Stable Solutions of Elliptic Partial Differential Equations[END_REF] and references therein.

Main results

Our first result establishes the non-existence of non-degenerate patterns in any unbounded convex domain.

Theorem 1.4 There exists no non-degenerate pattern to (1) in convex (possibly unbounded) domains.

This theorem has been proved by Dancer [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF] when Ω = R n . In this case, the result can be formally justified as follows: the stability of a given pattern in R n can be nothing but degenerate, since a continuum of patterns is given by translations of this same pattern. This argument is no longer valid if Ω is not invariant by translation.

Note that it is assumed in (1) that the solution has a bounded gradient. If we drop this assumption, then Theorem 1.4 do no longer apply, since u(x) := e x is a non-degenerate pattern of -u = -u in R.

The following result gives a classification of possibly degenerate patterns when the domain is convex and satisfies a growth condition at infinity.

Theorem 1.5 Assume that Ω is convex and satisfies

lim sup R→+∞ |Ω ∩ {|x| ≤ R}| R 2 < +∞. ( 5
)
Let u be a stable solution of (1).

1. If Ω is not a straight cylinder (i.e., Ω is not of the form R × ω, ω ⊂ R n-1 ), then u is constant.

2.

If Ω is a straight cylinder, then u is either constant or a monotonic flat solution. If, in addition, u is bounded then it connects two stable roots (z -, z + ) of f such that

z + z -f = 0.
If Ω satisfies (5), the above result ensures the non-existence of patterns if either Ω is not a straight cylinder, or

z2 z1 f = 0 for all z 1 = z 2 in {z ∈ R : f (z) = 0 and f (z) ≤ 0}.
Note that, in the special case of a straight cylinder Ω = R × ω, the convexity of the domain is degenerate in one direction. Planar patterns may indeed exist in such domains. For example, the Allen-Cahn equation in R, -u = u(1 -u 2 ) admits the explicit solution u : x → tanh x √ 2 which is stable (degenerate). This solution can be seen as a traveling wave connecting the two stable roots ±1, with speed c = 1 -1 f = 0. Condition ( 5) is usual in our context. It echoes with the celebrated Liouville type property of Berestycki, Caffarelli and Nirenberg (Theorem 1.7 in [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF]), of which a refined version is presented in Lemma 3.1. Under this assumption, we allow for instance convex domains that are subdomains of R 2 , or of the form R i × ω with ω bounded and i ∈ {1, 2}, or of the form

Ω = (x 1 , x ) ∈ R × R n-1 : x ∈ ω(x 1 ) ,
where for all x 1 ∈ R, ω(x 1 ) ⊂ R n-1 with lim sup According to De Giorgi's conjecture, it is reasonable to think that condition (5) can be substantially weakened, but not completely dropped since non-planar patterns may exist in R 8 . This problem has been widely investigated but many questions remain open, see section 3.2 for more details. Note, however, that straight cylinders (or the entire space) are domains for which convexity is degenerate. Up to the author's knowledge, not much is known about the classification of patterns in strictly convex domains.

Question 1 Can we relax assumption [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] under the assumption that the domain is strictly convex?

The conclusions, assumptions, and the proof of Theorem 1.5 are closely related to the work of Farina, Mari and Valdinoci [START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF]. In Theorem 1 there, the authors suppose the existence of a non-constant stable solutions of -∆u = f (u) with bounded gradient on a Riemannian manifold M without boundary and with nonnegative Ricci curvature. Without further assumptions on u, it is also assumed either that the manifold is parabolic (i.e. it does not admit any Green function) or that the domain satisfies a growth condition similar to (5) but replacing R 2 by R 2 log R (we prove that Theorem 1.5 actually holds under this weaker condition if the domain can be written as a cylinder with varying cross section, see the end of Section 3.2). Then, the authors in [START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF] prove that the manifold can be written M = R × N and that u only depends on the first variable. These two conclusions correspond respectively to the two conclusions of Theorem 1.5. Note that, since M is assumed to have no boundary, the only instance of M which is also a euclidian domain is the whole space R n . In section 3.4, we give further symmetry properties of patterns in unbounded domains, namely that patterns inherit the domain's invariance with respect to translations and planar rotations. As an application of our results, we propose to establish an asymptotic variant of Theorem 1.3. We consider a cylindrical domain which is asymptotically convex (see Figure 1), and show that any pattern converges to a constant. The following result deals with non-degenerate patterns.

Theorem 1. [START_REF] Barlow | On the Liouville Property for divergence form operators[END_REF] Let Ω ⊂ R N be a uniformly smooth cylindrical domain (with varying section) on the x 1 axis, which converges to a straight cylinder Ω ∞ := R × ω ∞ when x 1 → +∞ (Definition 4.2). Suppose that ω ∞ is convex, and let u be a bounded non-degenerate pattern of (1). Then, u(x 1 , •) converges C 2 loc to a stable root of f when x 1 → +∞.

The case of possibly degenerate patterns is treated in Theorem 4.1. Section 4.2 contains other symmetry results, if the limiting domain Ω ∞ is invariant under a translation or a rotation. In particular, we prove the following.

Corollary 1.7

Let Ω ⊂ R N be a cylindrical domain (with a variable section) on the x 1 axis, which converges to a straight cylinder Ω ∞ := R × ω ∞ , with w ∞ ⊂ R n-1 bounded, when x 1 → +∞ (Definition 4.2). Let u be a bounded stable non-degenerate solution of (1). Then, when

x 1 → +∞, u(x 1 , •) converges C 2 loc to u ∞ (•), which is a stable solution in the section ω ∞ : -∆ x u ∞ = f (u ∞ ) in ω ∞ , ∂ ν u ∞ = 0 on ∂ω ∞ . ( 6
)
The asymptotic symmetry results presented in Section 4.2 actually hold when the stability of the solution is only assumed to hold outside a compact set, see Remark 4.6.

Outline of the paper

Section 2 is devoted to some preliminaries and the classical proof of Theorem 1.3. We study the nonexistence of patterns in unbounded domains in section 3, that is, we prove Theorem 1.4, Theorem 1.5, and also recall classical further symmetry properties for patterns when the domain is invariant under translations or rotations.

In section 4 we establish asymptotic symmetries for patterns when the domain satisfies a geometrical property asymptotically. We prove Theorem 1.6 and give further properties.

In the Appendix A, we define and study the notion of generalized principal eigenvalue in unbounded domains. We also discuss the different definitions of stability in Appendix B, and the isolation of stable solutions in Appendix C.

Preliminaries: the classical case of bounded convex domains

To give an overview of the method and the difficulties arising when dealing with unbounded domains, we recall the classical proof of Theorem 1.3 which deals with bounded domains. It also allows us to introduce a key lemma. Let Ω be a smooth, convex, bounded domain, u a stable solution of (1) and set v i := ∂ xi u, for all i ∈ {1, . . . , n}.

Step 1. On the one hand, differentiating (1) with respect to x i , we find that v i := ∂ xi u satisfies the linearized equation

-∆v i -f (u)v i = 0 in Ω.
From an integration by part, we obtain

F(v i ) = ∂Ω v i ∂ ν v i = 1 2 ∂Ω ∂ ν v 2 i ,
with F from (2). On the other hand, as u is stable, we have F(•) ≥ 0 and

0 ≤ F(v i ) ≤ n k=1 F(v k ) = 1 2 ∂Ω ∂ ν |∇u| 2 . ( 7
)
(If Ω is unbounded, the computations are not licit and need to be adapted, see section 3.1.)

Step 2. When the domain is convex, the above integral turns out to be nonpositive, as stated in the following key lemma. This is where the convexity of the domain comes into play. It can be found in [START_REF] Casten | Instability results for reaction diffusion equations with Neumann boundary conditions[END_REF][START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF], but a simple proof is presented at the end of the section for completeness. Note that the lemma remains valid if the domain is unbounded.

Lemma 2.1 ( [20, 42])

Let Ω be a smooth convex domain. If u is a C 2 function such that

∂ ν u = 0 on ∂Ω, ( 8
)
then ∂ ν |∇u| 2 ≤ 0 on ∂Ω.
From the above lemma and the inequality (7), we conclude that for all i ∈ {1, . . . , n}, we have F(v i ) = 0. At this step, if u is not constant (i.e. v i ≡ 0 for some i), then we get λ 1 = 0, i.e., u is stable degenerate.

Step 3. Since F(v i ) = 0 and λ 1 ≥ 0, we deduce that v i minimizes F. It is then classical that v i is a multiple of the eigenfunction associated to λ 1 , denoted ϕ, which is unique (up to a multiplicative constant) and positive in Ω.

(This conclusion may fail in unbounded domains, see section 3.2.)

Step 4. From ∂ ν u = 0 on the closed surface ∂Ω, we deduce that v i vanishes on some point of the boundary. But as v i is colinear to ϕ, we conclude v i ≡ 0, which completes the proof.

(If Ω is a straight cylinder, the above conclusion fails and v i may be a nonzero multiple of ϕ.)

Before proving Lemma 2.1, we need the following definition.

Definition 2.2 Let Ω ⊂ R n . A "representation of the boundary" is a pair (ρ, U ) where ρ is a C 2 function defined on U a neighborhood of ∂Ω such that ρ(x)      < 0 if x ∈ Ω ∩ U = 0 if x ∈ ∂Ω > 0 if x ∈ U \Ω and ∇ρ(x) = ν(x) ∀x ∈ ∂Ω,
where ν(x) is the outer normal unit vector of ∂Ω at x.

It is classical that such a representation of the boundary always exists for C 2,1 domains, see e.g. section 6.2 of [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF].

Proof (of Lemma 2.1) Let us consider (ρ, U ) a representation of the boundary for Ω. Equation ( 8) becomes ∇u • ∇ρ = 0 on ∂Ω.

As ∇u is tangential to ∂Ω, we can differentiate the above equality with respect to the vector field ∇u. It gives, on ∂Ω,

0 = ∇ (∇u • ∇ρ) • ∇u = ∇u • ∇ 2 u • ∇ρ + ∇u • ∇ 2 ρ • ∇u.
From this, we infer

∂ ν |∇u| 2 = ∇ |∇u| 2 • ∇ρ = 2∇u • ∇ 2 u • ∇ρ = -2∇u • ∇ 2 ρ • ∇u,
Since Ω is convex, for all x 0 ∈ ∂Ω, we have that ∇ 2 ρ(x 0 ) is a nonnegative quadratic form in the tangent space of ∂Ω at x 0 . As ∇u is tangential to ∂Ω, we deduce from the above equation that ∂ ν |∇u| 2 is nonpositive.

In [START_REF] Casten | Instability results for reaction diffusion equations with Neumann boundary conditions[END_REF], the authors give the following remarkable geometrical interpretation of the above lemma. Consider a bounded convex domain Ω ⊂ R 2 . As u satisfies Neumann boundary conditions, its level set cross the border ∂Ω orthogonally. Since the domain is convex, these level sets go apart one from each other as we move outward ∂Ω. As |∇u| corresponds to the inverse of the distance of two level sets, it implies that |∇u| decreases as we move outward Ω, hence the result.

3 Patterns in unbounded domains

Non-degenerate patterns -proof of Theorem 1.4

The proof follows the same lines as the first two steps of section 2. But since Ω is unbounded, the computations which lead to "F(v i ) = 0" are not licit. We shall instead perform the computations on truncated functions. We denote v i := ∂ xi u for i ∈ {1, . . . , n}. Differentiating (1), we find that all the v i satisfy

v i (∆v i + f (u)v i ) ≥ 0 in Ω. ( 9
)
We consider a cut-off function: for R > 0, set

χ R (x) := χ |x| R , ∀x ∈ R n , ( 10
)
with χ a smooth nonnegative function such that

χ(z) = 1 if 0 ≤ z ≤ 1 0 if z ≥ 2 , |χ | ≤ 2.
Multiplying (9) by χ 2 R , integrating on Ω, using the divergence theorem and ( 15) we obtain

0 ≤ ∂Ω χ 2 R v i ∂ ν v - Ω ∇ χ 2 R v i • ∇v i + Ω f (u)χ 2 R v 2 i . Using the identity |∇ [χ R v i ]| 2 = ∇ χ 2 R v i • ∇v i + |∇χ R | 2 v 2
i and rearranging the terms, we deduce

λ 1 Ω χ 2 R v 2 i ≤ F (χ R v i ) ≤ 1 2 ∂Ω χ 2 R ∂ ν v 2 i + Ω |∇χ R | 2 v 2 i . ( 11
) Setting v = |∇u|, we have v 2 = v 2 1 + • • • + v n
and therefore summing the above inequality over i ∈ {1, . . . , n} gives

λ 1 Ω χ 2 R v 2 ≤ 1 2 ∂Ω χ 2 R ∂ ν v 2 + Ω |∇χ R | 2 v 2 .
Since Ω is convex, Lemma 2.1 implies that ∂ ν v 2 = ∂ ν |∇u| 2 ≤ 0 on ∂Ω. We deduce

λ 1 ≤ Ω |∇χ R | 2 v 2 Ω χ 2 R v 2 ≤ 4 R 2 Ω∩{|x|<2R} v 2 Ω∩{|x|<R} v 2 =: 4α R .
We claim that lim inf

R→+∞ α R ≤ 0. ( 12
)
If ( 12) holds, then λ 1 ≤ 0, which completes the proof. By contradiction, let us assume α R ≥ δ > 0. We use the notation C(R) := Ω∩{|x|<R} v 2 so that we can write α R = C(2R) R 2 C(R) . The contradictory assumptions now reads C(2R) ≥ δR 2 C(R). Iterating this inequality for a fixed R > 0, we find

C(2 j R) ≥ K δR 2 j for all integer j ≥ 1,
where, here and later, positive constants are generically denoted K. In addition, v is bounded, hence

C(R) ≤ KR n . We have K 2 j R n ≥ δR 2 j .
If R is large enough, we reach a contradiction as j goes to +∞. Thereby, we have proved [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed II -Cylindrical-type domains[END_REF], and the proof of Theorem 1.4 is complete.

A Liouville type result, or the simplicity of the principal eigenvalue

In this section, we adapt the step 3 of section 2 to unbounded domains. Let us first introduce the principal eigenfunction ϕ, associated to λ 1 . The Euler-Lagrange equation associated with the functional F is obtained as a linearization of (1) at u:

-∆ψ -f (u)ψ = 0, in Ω, ∂ ν ψ = 0 on ∂Ω. ( 13
)
If the domain Ω is bounded, then λ 1 is an eigenvalue of the linearized operator [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], called the principal eigenvalue. When the domain is unbounded, we will refer to λ 1 as the generalized principal eigenvalue. It is associated with an eigenfunction ϕ (Proposition A.2) which is positive on Ω and satisfies

-∆ϕ -f (u)ϕ = λ 1 ϕ in Ω, ∂ ν ϕ = 0 on ∂Ω.
The term generalized comes from the fact that ϕ may not belong to H 1 (Ω). See Appendix A for more details.

If the domain is bounded, then λ 1 is simple. This may no longer hold in unbounded domains. However, the following lemma claims that, if the domain satisfies [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF], any bounded minimizer of F is a multiple of ϕ.

It is a refinement of Theorem 1.7 in [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF]. [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] and let u be a stable solution of [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF].

Lemma 3.1 Let Ω ⊂ R n satisfy
Let v be a C 1 (Ω) ∩ L ∞ (Ω) satisfy v(∆v + f (u)v) ≥ 0 in Ω. ( 14
)
Consider the cut-off function introduced in (10) and assume that there exists a sequence of real positive numbers

(R n ) n≥1 diverging to +∞ such that ∂Ω χ 2 Rn ∂ ν v 2 ≤ 0, ∀n = 1, 2 . . . ( 15
)
Then v ≡ Cϕ for some constant C ∈ R, where ϕ is a principal eigenfunction associated with λ 1 .

Proof (Lemma 3.1) We follow the method of [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF]. Let us set σ = v ϕ and show that σ is constant. From ( 14), we deduce σϕ (ϕ∆σ + 2∇ϕ • ∇σ + σ (∆ϕ + f (u)ϕ)) ≥ 0, a.e. in Ω.

From λ 1 ≥ 0 and the equation satisfied by ϕ, we obtain σ∇ • (ϕ 2 ∇σ) ≥ 0, a.e. in Ω.

Multiplying by χ 2 R (defined in [START_REF] Berestycki | Nonlinear scalar field equations, I. Existence of a ground state[END_REF]), integrating on Ω and using the divergence theorem, we find

0 ≤ ∂Ω χ 2 R σϕ 2 ∂ ν σ - Ω ϕ 2 ∇ χ 2 R σ • ∇σ = ∂Ω χ 2 R σϕ 2 ∂ ν σ - Ω ϕ 2 χ 2 R |∇σ| 2 -2 Ω ϕ 2 χ R σ∇χ R • ∇σ.
Since ∂ ν ϕ = 0 on ∂Ω, the boundary term reads ∂Ω χ 2 R v∂ ν v, which is nonpositive from [START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF] when R = R n . From Cauchy-Schwarz inequality, we deduce

Ω χ 2 Rn ϕ 2 |∇σ| 2 ≤ 2 Ω 2Rn \Ω Rn χ 2 Rn ϕ 2 |∇σ| 2 Ω v 2 |∇χ Rn | 2 , ( 16
)
where Ω R := Ω ∩ {|x| < R}.

Recalling that χ R (x) := χ |x| R with |χ | ≤ 2, assumption (5) implies Ω v 2 |∇χ R | 2 is bounded, uniformly in R ≥ 1. (17)
From ( 16), we deduce that Ω χ 2 Rn ϕ 2 |∇σ| 2 is uniformly bounded. Using ( 16) again, we infer that it converges to 0 as R n → ∞. At the limit, we find Ω ϕ 2 |∇σ| 2 ≤ 0. Hence ∇σ = 0, which ends the proof.

The cornerstone of the proof is that σ∇ • (ϕ 2 ∇σ) ≥ 0 implies ∇σ = 0, where σ := v ϕ . The litterature refers to this property as a Liouville property. Originally introduced in [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF], it has been extensively discussed (see [START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in Rˆ3 and a conjecture of De Giorgi[END_REF][START_REF] Barlow | The Liouville property and a conjecture of De Giorgi[END_REF][START_REF] Gazzola | The sharp exponent for a Liouville-type theorem for an elliptic inequality[END_REF][START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF][START_REF] Moschini | New Liouville theorems for linear second order degenerate elliptic equations in divergence form[END_REF]) and used to derive numerous results (e.g. [START_REF] Alberti | On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property[END_REF][START_REF] Berestycki | One-dimensional symmetry of bounded entire solutions of some elliptic equations[END_REF][START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF][START_REF] Cabré | On the stability of radial solutions of semilinear elliptic equations in all of Rˆn[END_REF][START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF][START_REF] Dupaigne | Stable solutions of -\Delta u = f(u) in \Rˆn[END_REF]), in particular to prove the De Giorgi's conjecture in low dimensions. Lemma 3.1 is a refinement of this property for domains with a boundary, instead of Ω = R n . This is why we need the boundary condition [START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF]. This is the only step where (5) is needed, and it is thus a natural question to ask if this assumption can be relaxed. In the proof, ( 5) is used to derive [START_REF] Cabre | Stable solutions to semilinear elliptic equations are smooth up to dimension 9[END_REF], thus the choice of χ R seems crucial. However, in [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF], the authors consider the optimal χ R by taking the capacitary test function (see e.g. [START_REF] Helms | Introduction to potential theory[END_REF]), i.e., a solution of the minimization problem inf

χ∈H 1 (R 2 ) R≤|x|≤R |∇χ(x)| 2 dx, ξ(x) = 1 if |x| ≤ R 0 if |x| ≥ R . ( 18
)
That, in fact, does not allow to substantially relax condition [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]. In [START_REF] Barlow | On the Liouville Property for divergence form operators[END_REF], Barlow uses a probabilistic approach to establish that the aforementioned Liouville property (and consequently Lemma 3.1) does not hold in Ω = R n , n ≥ 3. It is thus reasonable to think that condition (5) cannot be relaxed, yet this is an open question. We also cite [START_REF] Gazzola | The sharp exponent for a Liouville-type theorem for an elliptic inequality[END_REF], in which ( 5) is proved to be sharp, however, we point out that, there, the condition v ∈ L ∞ is not satisfied. Note also that, in the present work, we only apply Lemma 3.1 to functions v, which are derivatives of u, which is a stronger condition than [START_REF] Bombieri | Minimal cones and the Bernstein Problem[END_REF]. In this context, not much is known about whether (5) could be relaxed. Indeed, up to the author's knowledge, the only available counterexample is for Ω = R n , n ≥ 7 [START_REF] Pacard | Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones[END_REF][START_REF] Pino | On De Giorgi's conjecture in dimension N\ge 9[END_REF]. However, we can sometimes relax (5) under further assumptions, either on Ω, f , v, or ϕ. From a remark in [START_REF] Dupaigne | Stable solutions of -\Delta u = f(u) in \Rˆn[END_REF], if f ≥ 0, we can relax [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]

to sup R→+∞ |Ω∩{|x|≤R}| R 4
< +∞. We can also adapt the arguments of [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF][START_REF] Cabré | On the stability of radial solutions of semilinear elliptic equations in all of Rˆn[END_REF][START_REF] Moschini | New Liouville theorems for linear second order degenerate elliptic equations in divergence form[END_REF] to show that assumption (5) can be replaced by v ∈ H 1 (Ω), or v = o(|x| 1-n 2 ), or inf Ω ϕ > 0. In addition, we point out that Lemma 3.1 holds for a large class of domains satisfying sup

R→+∞ |Ω ∩ {|x| ≤ R}| R 2 ln R < +∞.
Namely, let Ω be of the form

Ω := (x, x ) ∈ R 2 × R n : x ∈ ω(x) ,
where, ∀x ∈ R n , ω(x) ⊂ R n is bounded and

sup |x|→+∞ |ω(x)| ln |x| < +∞.
Then, to show [START_REF] Cabre | Stable solutions to semilinear elliptic equations are smooth up to dimension 9[END_REF], we use the cut-off

χ R (x) =          1 if |x| ≤ R, ln R 2 -ln |x| ln R 2 -ln R if R ≤ |x| ≤ R 2 , 0 if |x| ≥ R 2 .
This cut-off was first introduced in [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF] as a solution of [START_REF] Caffarelli | Uniform convergence of a singular perturbation problem[END_REF] for n = 2.

Non-existence of patterns -proof of Theorem 1.5

We prove Theorem 1.5. Let u be a stable solution of (1) and consider ϕ a principal eigenfunction associated with λ 1 . As a consequence of Lemma 3.1 and Lemma 2.1, we have the following intermediate result.

Lemma 3.2 For any

ξ ∈ R n , ∇u • ξ ≡ C ξ ϕ for some constant C ξ .
Proof We assume without loss of generality that |ξ| = 1, and that ξ coincide with e 1 for (e 1 , . . . , e n ) an orthonormal basis of R n . We set

v i := ∂ xi u for i ∈ {1, • • • , n}.
Differentiating (1), we find that v i satisfies [START_REF] Bombieri | Minimal cones and the Bernstein Problem[END_REF]. Moreover, we know by assumption that the v i are bounded. Now, we show that all the v i satisfy [START_REF] Cabré | A Conjecture of De Giorgi on Symmetry for Elliptic Equations in R[END_REF]. On the one hand, since Ω is convex, Lemma 2.1 implies

n i=1 ∂Ω χ 2 R ∂ ν v 2 i = ∂Ω χ 2 R ∂ ν |∇u| 2 ≤ 0.
On the other hand, Lemma 3.1 implies in particular that for any i ∈ {1, . . . , n} we have ∂Ω χ 2 R ∂ ν v 2 i ≥ 0 for all R 1, i.e., for R large enough, all the terms of the above sum are nonnegative. As the sum is nonpositive, all the terms must be zero. Then, we apply Lemma 3.1 to conclude. Let us complete the proof of Theorem 1.5. The mapping

R n → R ξ → C ξ
is linear and therefore vanishes on a hyperplane H. Setting e ∈ S n-1 the unit vector orthogonal to H, we have ∇u = |∇u|e, hence u varies only in the direction e (in other words, u is a function of only one scalar variable).

If Ω is not straight in the direction e, there exists a point x 0 ∈ ∂Ω on which the outer normal derivative is not colinear with e. From ∂ ν u = 0 on ∂Ω, we deduce ∂ e u(x 0 ) = 0. From ϕ > 0 on Ω, we deduce ∂ e u ≡ 0, thus u is constant.

If Ω is straight in the direction e, then ∂ e u may be a nonzero multiple of ϕ. To fix ideas, we assume that e corresponds to the x 1 direction. Since v 1 ≡ C 1 ϕ, it is of constant sign, hence u is flat and monotic.

Let us now assume that u is bounded. First, note that, since u is monotonic, it has a limit z + when x 1 → +∞. Setting u n (x 1 ) = u(x 1 + n) and using classical elliptic estimates, we can extract a subsequence that C 2 loc -converges to a stable solution u ∞ of (1) (See section 4 for more details. Note also that Ω is invariant under translation in the x 1 direction). From u ∞ ≡ z + , we deduce that z + must be a stable root of f . Identically, when x 1 → -∞, u converges to a stable root of f , denoted z -.

If z + = z -, then u is constant. Let us assume z -= z + , and fix M > 0. Multiplying -u = f (u) by u and integrating on

x 1 ∈ [-M, M ] gives 1 2 u (-M ) 2 -u (M ) 2 = u(M ) u(-M ) f.
As u (±∞) = 0 (indeed, u is integrable and u is bounded), when M goes to +∞ we obtain z + z -f = 0. The proof of Theorem 1.5 is thereby complete.

Further symmetries

In this section, we establish further symmetries for patterns in unbounded domains. Namely, we shall see that patterns inherit from the domain's invariance with respect to translations and planar rotations. The results of this section are mostly classical, except for some minor modifications. We provide some proofs for completeness.

The following proposition deals with cylinders, possibly not convex, which are straight in some directions. This result is somehow already contained in [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF][START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF][START_REF] Farina | Stable Solutions of Elliptic Equations on Riemannian Manifolds[END_REF][START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF].

Proposition 3.3 Let Ω = R n × ω with ω ⊂ R m bounded
and let u be a stable solution of (1). For x ∈ Ω, we generically denote x = (x 1 , . . . , x n , x 1 , . . . , x m ).

1. If u is stable non-degenerate, then u does not depend on (x 1 , . . . , x n ).

If u is stable and n

= 1, then u is monotonic with respect to x 1 .
3. If u is stable and n = 2, the dependance of u with respect to (x 1 , x 2 ) occurs through a single scalar variable x 0 ∈ R. Moreover, u is monotonic with respect to x 0 .

Proof (Proposition 3.3) We set v i := ∂ xi u for i ∈ {1, . . . , n}, which are bounded by assumption. Differentiating (1), we find that v i satisfies [START_REF] Berestycki | One-dimensional symmetry of bounded entire solutions of some elliptic equations[END_REF]. We can then proceed as in the proof of Theorem 1.4 to show that (11) holds, namely

λ 1 Ω χ 2 R v 2 i ≤ F (χ R v i ) ≤ 1 2 ∂Ω χ 2 R ∂ ν v 2 i + Ω |∇χ R | 2 v 2 i ,
where χ R is the cut-off function introducedin [START_REF] Berestycki | Nonlinear scalar field equations, I. Existence of a ground state[END_REF]. Since Ω is straight in the directions x i , we have ∂ ν v i = 0 on ∂Ω. Therefore we have

λ 1 Ω χ 2 R v 2 i ≤ Ω |∇χ R | 2 v 2 i
and we can conclude as in the proof of Theorem 1.4 (in particular using [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed II -Cylindrical-type domains[END_REF]) that if v i ≡ 0 then λ 1 ≤ 0. It proves the first astatement. Next, if n ≤ 2 then the domain satisfies [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]. Since v i satisfies ( 14) and ( 15), we deduce from Lemma 3.1 that v i is a multiple of ϕ. The second assertion follows from the fact that ϕ > 0. To prove the last statement, note that, as in the proof of Theorem 1.5, there exists a direction ξ ∈ S n-1 for which ∇u • ξ ≡ 0.

We are now interested in cylinders which are invariant with respect to a planar rotation and shall see that patterns inherit this symmetry. This property was initially stated in [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF] for bounded domains, and has been extended to manifolds [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF][START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF][START_REF] Farina | Stable Solutions of Elliptic Equations on Riemannian Manifolds[END_REF][START_REF] Wolansky | Instability results for reaction diffusion equations over surfaces of revolutions[END_REF][START_REF] Jimbo | On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions[END_REF][START_REF] Lopes | Radial symmetry of minimizers for some translation and -rotation invariant functionals[END_REF][START_REF] Punzo | The existence of patterns on surfaces of revolution without boundary[END_REF].

Definition 3.4 A domain Ω ⊂ R n+2 is said to be θ-invariant if Ω = Ω × [0, 2π), where Ω ⊂ R n × R + in some cylindrical coordinates (x, r, θ) ∈ R n × R + × [0, 2π).
When considering a θ-invariant domain Ω, we further assume that the radial section is uniformly bounded: sup

(x,r)∈Ω r < +∞. ( 19
)
In particular, it guarantees that if u is a solution of (1) in Ω, then ∂ θ u is bounded. Note that this assumption is strictly needed, since planar patterns may exist in R 2 .

Lemma 4.3

We can extend u to a uniformly C |x -y| > 0.

The extended function, denoted ũ, satsfies an elliptic equation on U for which classical global C 2,α estimate hold (see e.g. Theorem 6.30 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]). This procedure is classical but technical, see for instance Appendix A in [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed II -Cylindrical-type domains[END_REF]. From this, we infer global C 2,α estimates for ũ in Ω. Finally, choosing any open set Ũ such that Ω ⊂ Ũ ⊂ U , we can define ũ ∈ C 2,α (R n \Ω) which coincides with ũ in Ũ and is identically 0 in R N \U .

We consider the ω-limit set

Σ u := y∈R {u[y ] : y ≥ y} ⊂ L ∞ loc (R N ), ( 20
)
where

u[y] : (x 1 , . . . , x N ) → u(x 1 + y, . . . , x N ), ∀y ∈ R.
The topological closure should be understood in the L ∞ loc sense. The key point is the following observation, which states that a solution is "more stable at infinity".

Lemma 4.4 Let u ∞ ∈ Σ u . It is a solution of (1) in Ω ∞ . Moreover, λ 1 (u, Ω) ≤ λ 1 (u ∞ , Ω ∞ ).
Proof Let u ∞ ∈ Σ u . There exists a sequence y n → +∞ such that u[y n ] → u ∞ . As a consequence of the locally uniform C 2,α estimates in Ω (see the proof of Lemma 4.3), we deduce that the convergence

u[y n ] → u ∞ occurs in C 2 loc . Thus, u ∞ is a solution of (1) in Ω ∞ . Consider ϕ ∈ C 2 (Ω)
-∆ϕ n -f (u[y n ])ϕ n = λ 1 (u, Ω)ϕ n in Ω[y n ], ∂ ν ϕ n = 0 on ∂Ω[y n ].
We can extend by reflexion ϕ n on a neighborhood of Ω[y n ] on which it satisfies an elliptic equation, thus can apply the Harnack inequality in Ω ∞ to infer that ϕ n is locally bounded, uniformly in n (for more details, see the proof of Proposition 1 p.30 in [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed II -Cylindrical-type domains[END_REF]). Thanks to classical elliptic estimates, we can extract a subsequence (still denoted n) such that ϕ n converges in

C 2 loc (Ω ∞ ) to some ϕ ∞ . Then, ϕ ∞ > 0 in Ω ∞ and (∆ + f (u ∞ ) + λ 1 (u, Ω))ϕ ∞ ≤ 0. From Corollary A.3, we deduce λ 1 (u ∞ , Ω ∞ ) ≥ λ 1 (u, Ω).
We also need the following general lemma, which is adapted from a classical result (see, for example, Theorem 2.9 in [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF]).

Lemma 4.5 Σ u is a connected set in the L ∞ loc (R n ) topology.
Proof By contradiction, assume there exists

Y ⊂ Σ u both open and closed, Y = ∅ and Y = Σ u . Note that Σ u is a compact subset of L ∞ loc (R n ), thus Y is compact. As, in addition, Σ u \Y is a closed set, there exists a an open set V ⊂ L ∞ loc (R n ) and a closed set F ⊂ L ∞ loc (R n ) such that Y V F and Y = Σ u ∩ F . Since Y = ∅ and Y = Σ u ,
there exists a real sequence (y n ) n≥0 such that y n → +∞ and, for all integer n ≥ 0,

u[y 2n ] ∈ V, u[y 2n+1 ] ∈ F.
By continuity of the mapping y → u[y], we deduce that, for all n ≥ 0, there exists ỹn

∈ [y 2n , y 2n+1 ] such that u[ỹ n ] ∈ F \V.
As the sequence u[ỹ n ] is uniformly C 2,α , it converges up to an extraction to some u ∞ ∈ Σ u . But we also have u ∞ ∈ F \V , thus u ∞ ∈ Y and u ∞ ∈ Y : contradiction.

We are now ready to prove the main results of this section.

Proof (Theorem 1.6 and Theorem 4.1) Let u ∞ ∈ Σ u . From Lemma 4.4, u ∞ is a stable solution of [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF] in Ω ∞ , which is convex. From Theorem 1.4 and Theorem 1.5, we deduce that u ∞ is constant. We have

Σ u ⊂ Z := {z ∈ R : f (z) = 0, f (z) ≤ 0} .
Assume f has only isolated zeros. Then Σ u is a discrete set, and is also connected (Lemma 4.5). Hence, it is a singleton, which achieves the proof.

Assume instead that u is stable non-degenerate, then we have

Σ u ⊂ Z := {z ∈ Z : f (z) < 0} ,
which is a discrete set, and we conclude as above.

Remark 4.6

The proof of Theorem 1.6 and Theorem 4.1 and also the proof of the other asymptotic symmetry properties presented in the next section can be adapted to the case where u is only assumed to be stable outside a compact K ⊂ Ω, i.e., when the condition on the sign of λ 1 is replaced by a condition on the sign of

λ 1,K = inf ψ∈C 1 0 (Ω\K) ψ L 2 =1 Ω\K |∇ψ| 2 -f (u)ψ 2 .
To adapt the proofs to this case, simply note that if Ω converges to Ω ∞ , then so does Ω \ K.

Further asymptotic symmetries

In this section, we establish further asymptotic symmetries for patterns when the domain is asymptotically straight in one direction or invariant with respect to a planar rotation. We define what we mean for a domain to satisfy a geometrical property asymptotically. Note that we allow the domain not to converge to a limiting domain (in the sense of Definition 4.2).

Definition 4.7

Let Ω ⊂ R n be a uniformly C 2,α domain. For any real sequence y n → +∞, Ω[y n ] converges (up to an extraction) to some Ω ∞ ⊂ R n (in the sense of Definition 4.2). We define the set of all possible limiting domains

Γ Ω := {Ω ∞ ⊂ R n : ∃y n → +∞, Ω[y n ] converges to Ω ∞ } .
We say that a domain Ω satisfies a geometrical property asymptotically if every Ω ∞ ∈ Γ Ω satisfies this property.

Proposition 4.8 Assume that Ω is asymptotically straight in a direction e ∈ S N . Let u be a stable solution of (1). If u is stable non-degenerate, then ∂ e u converges to 0, C 1 loc -uniformly when x 1 → +∞.

Proof Let y n → +∞. Up to an extraction (still denoted y n ), u[y n ] converges to some u ∞ ∈ Σ u in C 2,α loc and Ω[y n ] converges to some Ω ∞ ∈ Γ Ω . From Lemma 4.4, we deduce that u ∞ is a stable non-degenerate solution of (1) in Ω ∞ .
As Ω ∞ is straight in the direction e, using Proposition 3.3 we deduce that ∂ e u ∞ ≡ 0, hence the result.

Corollary 1.7 follows from the previous result in the particular case when the direction e coincides with that of x 1 .

Proof (Corollary 1.7) We apply the same method as in the proof of Theorem 1.6, using Proposition 3.3 and the fact that stable non-degenerate solutions are isolated among solutions (Lemma C.1).

Note that, if we assume that the set of stable solutions of ( 6) is discrete, then Corollary 1.7 extends to possibly degenerate patterns.

We now turn to the case of a domain which is asymptotically invariant with respect to a planar rotation. Let u be a stable solution of (1). Assume either that u is stable non-degenerate, or that Ω satisfies [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]. Then ∂ θ u → 0, C 1 loc -uniformly when x 1 → +∞.

Proof We proceed as in the proof of Proposition 4.8 and we use Proposition 3.5 instead of Proposition 3.3.

A Generalized principal eigenvalue

This section is devoted to defining the generalized principal eigenvalue of a linear operator, and to state some properties. The term generalized is used when dealing with unbounded domains, in which there may not exist eigenfunctions in H 1 . Here, we focus on the essential aspects and omit the details: the content of this section will be developed in a forthcoming paper [START_REF] Nordmann | The Maximum Principle for general boundary conditions in unbounded domains[END_REF] A.

Definition

We generally consider a smooth domain Ω and a linear elliptic operator

Lu(x) := div (A(x) • ∇u(x)) + B(x) • ∇u(x) + c(x)u(x), ∀x ∈ Ω, (21)
where, c : Ω → R, B : Ω →∈ R n , and

A : Ω → R n×n such that A(x) is positive-definite (uniformly in x ∈ Ω).
For simplicity, we assume that the coefficients are smooth. We associate the operator L with Neumann boundary conditions

Bu(x) := ∂ ν A u(x) = 0, ∀x ∈ ∂Ω,
with ν the outer normal derivative and ∂ ν A u := ν • A • ∇u the co-normal outer derivative of u associated with A. We focus here on Neumann boundary conditions, but we keep the notation B to emphasize that our statements can be adapted to other boundary conditions. We consider the following eigenproblem:

-Lψ = λψ in Ω,

Bψ = 0 on ∂Ω. ( 22
)
If the domain is bounded, the Krein-Rutman theory gives the existence of an eigenvalue λ 1 to [START_REF] Giorgi | Una estensione del teorema di Bernstein[END_REF], called the principal eigenvalue. This eigenvalue is real and minimizes the real part of the spectrum. In addition, λ 1 is simple, and is the only eigenvalue associated with a positive eigenfunction (called principal eigenfunction).

We let the reader refer to [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF][START_REF] Protter | Maximum Principles in Differential Equations[END_REF] for more details. We point out that a fundamental property is that the validity of the Maximum Principle for the operator (L, B) is equivalent to the condition λ 1 > 0.

If the domain is unbounded, Krein-Rutman's theory cannot be applied because the elliptic operator does not have compact resolvents. However, we can still define the notion of principal eigenvalue. Following the approach of [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF][START_REF] Rossi | Stability analysis for semilinear parabolic problems in general unbounded domains[END_REF], we give the following definitions.

Definition A.1 • A function u ∈ C 2 (Ω) is said to be a subsolution (resp. supersolution) if -Lu ≤ 0 (resp. ≥ 0) in Ω,
Bu ≤ 0 (resp. ≥ 0) on ∂Ω.

• We define the generalized principal eigenvalue of (L, B) as

λ 1 := sup {λ ∈ R : (L + λ, B) admits a positive supersolution} . ( 23
)
This definition coincides with the classical definition in the case of a bounded domain, and coincide with the definition (2) when L is self-adjoint. We are about to see that λ 1 admits a positive eigenfunction. However, λ 1 may not be simple.

A.2 Existence of a positive eigenfunction

A remarkable property is that, even in unbounded domains, λ 1 is associated with a positive eigenfunction.

Proposition A. [START_REF] Alberti | On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property[END_REF] Let Ω ⊂ R n be a smooth (possibly unbounded) domain and L an elliptic operator as in [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF]. There exists ϕ ∈ C 2 (Ω) which is positive on Ω and satisfies

-Lϕ = λ 1 ϕ in Ω, Bϕ = 0 on ∂Ω. ( 24
)
We refer to ϕ as a principal eigenfunction of (L, B).

Proof The proof follows closely [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] (see the proofs of Theorem 3.1 and Proposition 1). For any R > 0, let

D R ⊂ R n be a smooth connected open set such that B R ⊂ D R ⊂ B 2R
, with B R the ball of radius R. We also choose D R to be increasing with R, and to be such that From the results of Liberman [START_REF] Lieberman | Mixed boundary value problems for elliptic and parabolic differential equations of second order[END_REF], we know that all classical results (Schauder estimate, Maximum Principle, solvability, etc.) hold from the mixed boundary value problem above. As Ω R is bounded, the weak Krein-Rutman theorem (e.g., Corollary 2.2 in [START_REF] Nussbaum | Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem[END_REF]) provides a pair of principal eigenelements (λ R 1 , ϕ R ), where ϕ R ∈ W 2,n . From Hopf's lemma, we have ϕ > 0 on Ω. We choose the normalization ϕ(0) = 1. Note that we impose Dirichlet boundary conditions on Ω ∩ ∂D R to ensure the decreasing monotonicity of R → λ R 1 . Hence, λ R 1 converge to some λ 1 when R → +∞. Now, fix a compact 0 ∈ K ⊂ Ω and assume that R is large enough so that K ⊂ Ω R \D R . From Theorem 3.3 in [START_REF] Lieberman | Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations[END_REF] and Theorem 4.3 in [START_REF] Lieberman | Pointwise estimates for oblique derivative problems in nonsmooth domains[END_REF], we derive a Harnack estimate, that is,

∂D R ∩ ∂Ω is a C 2 (n -2)-dimensional manifold. Set Ω R := Ω ∩ D R
sup K ϕ R ≤ C inf K ϕ R
with a constant C independent of R. From ϕ(0) = 1, we deduce that ϕ R is bounded in K, uniformly in R. From classical Schauder estimates, we deduce that ϕ R is C 2,α (K), uniformly in R. Up to extraction of a subsequence, ϕ R converges to some ϕ in C 2 (K). From a diagonal argument, we are provided with ϕ ∈ C 2 (Ω) which satisfies

-Lϕ = λ 1 ϕ in Ω, Bϕ = 0 on ∂Ω,
and ϕ > 0 on Ω. Consequently λ 1 = λ 1 , which achieves the proof.

As a direct consequence, Corollary A. [START_REF] Alikakos | On the singular limit in a phase field model of phase transitions[END_REF] There exists a positive supersolution of (L, B) in Ω if and only if λ 1 ≥ 0.

A.3 The Rayleigh-Ritz variational formula

In the self-adjoint case, i.e., if B ≡ 0 in [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF], the principal eigenvalue can be expressed through the Rayleigh-Ritz variational formula. This result is classical in bounded domains.

Proposition A. [START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in Rˆ3 and a conjecture of De Giorgi[END_REF] Assume Ω is smooth (possibly unbounded) and that L is a self-adjoint elliptic operator.

For λ 1 defined in [START_REF] Del Pino | A counterexample to a conjecture by De Giorgi in large dimensions[END_REF], we have

λ 1 = inf ψ∈H 1 (Ω) ψ L 2 =1 F(ψ) := inf ψ∈H 1 (Ω) ψ L 2 =1 Ω |∇ψ| 2 A -cψ 2 , ( 25
)
with |∇ψ| 2 A := ∇ψ • A • ∇ψ.
Note that since the coefficient c is bounded, from the dominated convergence theorem we deduce that the infimum in ( 25) can be taken equivalently on test functions ψ ∈ C 1 0 (Ω).

Proof From the dominated convergence theorem and classical density results, it is equivalent to take the infimum on compactly supported smooth test functions in [START_REF] Dupaigne | Stable solutions of -\Delta u = f(u) in \Rˆn[END_REF], namely

λ 1 = inf ψ∈C 1 c (Ω) ψ L 2 =1 F(ψ),
where C 1 c (Ω) is the space of continuously differentiable functions with compact support in Ω. The remaining of the proof is classical and can be adapted from the proof of Proposition 2.2 (iv) in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] (which itself relies on [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF][START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]).

A.4 The Maximum Principle

We give, as a complement, some results on the link between the sign of λ 1 and the validity of the Maximum Principle in unbounded domains. The proofs of the following statements are underlying in the content of the present article, and we leave the details to a forthcoming note [START_REF] Nordmann | The Maximum Principle for general boundary conditions in unbounded domains[END_REF]. Definition A. [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] We say that (L, B) satisfies the Maximum Principle if any subsolution with finite supremum is nonpositive.

For simplicity, we focus on self-adjoint operators, that is, we assume B ≡ 0 in [START_REF] Dancer | Stable and Finite Morse Index solutions on Rˆn or on bounded domains with small diffusion[END_REF]. We can then express λ 1 through the Rayleigh-Ritz variational formula [START_REF] Dupaigne | Stable solutions of -\Delta u = f(u) in \Rˆn[END_REF].

The first result states that the (strict) sign of λ 1 is equivalent to the validity of the Maximum Principle.

Proposition A.6 Assume L is self-adjoint.

1. If λ 1 > 0, (L, B) satisfies the Maximum Principle.

2. If λ 1 < 0, (L, B) does not satisfy the Maximum Principle.

No general answer holds for the degenerate case λ 1 = 0. Nevertheless, the following proposition states the validity of what could be called a Critical Maximum Principle when λ 1 ≥ 0 if the domain satisfies a growth condition at infinity.

Proposition A.7 Suppose that L is self-adjoint and that the domain Ω satisfies [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]. Let ϕ be an eigenfunction associated with λ 1 . If λ 1 ≥ 0, then a subsolution with finite supremum is either nonpositive or a multiple of ϕ.

A first consequence of this result is the simplicity of λ 1 if it admits a bounded eigenfunction.

Corollary A.8 Under the same conditions, if λ 1 admits a bounded eigenfunction, then λ 1 is simple.

The simplicity of λ 1 should be understood as follows: if ψ ∈ C 2 (Ω) is a solution of ( 24), then it is a scalar multiple of ϕ.

We also give the following necessary and sufficient condition for the validity of the Maximum Principle in the critical case λ 1 = 0. Corollary A.9 Under the same conditions, further assume λ 1 = 0, and let ϕ be an associated eigenfunction. Then, (L, B) satisfies the Maximum Principle if and only if ϕ is not bounded.

B On the different definitions of stability

When considering stability from a dynamical point of view, one can come up with the two following definitions.

Definition B.1

A solution u of (1) is said to be dynamically stable if, given any ε > 0, there exists δ 0 > 0 such that for any v 0

(x) with v 0 -u L ∞ ≤ δ 0 , we have v(t, •) -u(•) L ∞ ≤ ε, ∀t > 0,
where v(t, x) is the solution of the evolution problem

     ∂ t v(t, x) -∆v(t, x) = f (v(t, x)) ∀x ∈ Ω, ∀t > 0, ∂ ν v(t, x) = 0 ∀x ∈ ∂Ω, ∀t > 0, v(t = 0, x) = v 0 (x) ∀x ∈ Ω. ( 26
)
Definition B.2 A solution u of (1) is said to be asymptotically stable if there exists δ 0 > 0 such that for any

v 0 (x) with v 0 -u L ∞ ≤ δ 0 , we have v(t, •) -u(•) L ∞ → 0, when t → +∞,
where v(t, x) is the solution of [START_REF] Farina | Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds[END_REF].

The following proposition clarifies the hierarchy of the different definitions of stability.

Proposition B.3

Let u be a solution of (1) and λ 1 from (2). The following implications hold u asymptotically stable ⇒ u dynamically stable ⇒ λ 1 ≥ 0.

Proof The first implication is trivial. Let us show the second implication by contradiction: assume λ 1 < 0 and that u is dynamically stable. For R > 0, define the truncated domain Ω R := Ω ∩ {|x| < R} and consider the following mixed-boundary eigenvalue problem:

find λ 1,R ∈ R and ϕ R ∈ C 2 (Ω R ) satisfying      -∆ϕ R -f (u)ϕ R = λ 1,R ϕ R in Ω R , ∂ ν ϕ R = 0 on ∂Ω ∩ {|x| < R}, ϕ R = 0 on Ω ∩ {|x| = R}. ( 27
)
From a recent result of Rossi [52, Theorem 2.1], we know that, for almost every R > 0, the eigenproblem (27) admits a unique eigenpair (λ 1,R , ϕ R ) such that ϕ R > 0 on Ω ∩ {|x| < R}. Moreover, R → λ 1,R is strictly decreasing and lim R→+∞ λ 1,R = λ 1 . Let us fix R > 0 large enough such that λ 1,R < 0. We also choose the normalization ϕ R L ∞ = 1. Let us consider a parameter ε > 0 small enough such that

η ε := sup ũ∈[inf Ω R u,sup Ω R u] |h|≤ε f (ũ) - f (ũ + h) -f (ũ) h < -λ 1,R , (28)
and δ 0 given by Definition B.1. Consider v the solution of the parabolic equation ( 26) with initial datum v 0 := u + δ 0 ϕ R , and set h(t, x) = v(t, x) -u(x). On the one hand, since v 0 -u L ∞ ≤ δ 0 , the stability assumption implies h(t, •) L ∞ ≤ ε for all time t ≥ 0. On the other hand, h satisfies

     ∂ t h(t, x) -∆h(t, x) ≥ (f (u(x)) -η ε ) h(t, x) in Ω R , ∂ ν h = 0 on ∂Ω ∩ {|x| < R}, h ≥ 0 on Ω ∩ {|x| = R}
From the parabolic comparison principle, we infer h(t, x) ≥ h(t, x) := e -(λ1+ηε)t δ 0 ϕ R (x) for all t ≥ 0 and x ∈ Ω R . Using that λ 1 + η δ < 0, we deduce that h(t, •) L ∞ diverges to +∞ when t becomes large: contradiction.

Remark B.4

Note that, in the proof, the perturbation δ 0 ϕ R has a compact support in Ω and an arbitrarily small L ∞ norm. Thus, if λ 1 < 0 then (26) drives u + h away from u for any h which is positive or negative on Ω R if R is large enough.

One can ask whether the following implication holds:

λ 1 > 0 ⇒ u asymptotically stable. ( 29
)
This implication is classical when the domain is bounded (see Proposition 1.4.1 in [START_REF] Dupaigne | Stable Solutions of Elliptic Partial Differential Equations[END_REF]), but it is not clear whether it extends to unbounded domains.

Question 2 Does (29) hold in unbounded domains ?

We think that, in general, the answer is negative. Nevertheless, as a consequence of the results of the present paper, we give a positive answer for unbounded convex domains.

Proposition B. [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] Let Ω ⊂ R n be a smooth convex domain (possibly unbounded) and u be a solution of [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF]. Then λ 1 > 0 ⇒ u asymptotically stable ⇒ u dynamically stable ⇒ λ 1 ≥ 0.

Proof From Proposition B.3, we only have to show the first implication. Assume λ 1 > 0. We deduce from Theorem 1.4 that u is constant. Thus λ 1 = -f (u) and ϕ is constant. We choose ε small enough such that η ε ∈ (0, λ1 2 ) with η ε defined in [START_REF] Farina | The state of the art for a conjecture of De Giorgi and related problems[END_REF]. Let v 0 be as in Definition B.2 (we use the same notations). We set T := sup{t > 0 : h(t, •) L ∞ ≤ ε}. By continuity and the choice of v 0 , we know that T > 0. We have ∂ t h(t, x) -∆h(t, x) ≤ (f (u) + η ε ) h(t, x) ∀t ∈ (0, T ), x ∈ Ω, ∂ ν h(t, x) = 0 ∀t ∈ (0, T ), x ∈ ∂Ω.

From f (u) + η ε ≤ -λ1 2 and the parabolic comparison principle, we obtain h(t, •) L ∞ ≤ εe -λ 1 2 t , for all t ∈ (0, T ). We deduce T = +∞ and h(t, •) L ∞ → 0 when t → +∞, thus u is asymptotically stable.

C Isolation of stable solutions

We give a brief discussion on the isolation of stable solutions of (1) in the set of all solutions. Note that this question is crucial in the proof of Theorem 1.6 and Theorem 4.1, since the key point is to show that Σ u is a discrete set. When considering the L ∞ topology, we have the following.

Lemma C.1

Let Ω ⊂ R n be a smooth domain (possibly unbounded), and denote S the set of solutions of (1) in Ω. Let u ∈ S be either stable non-degenerate, or unstable non-degenerate (i.e. λ 1 < 0). Then, u is isolated in S for the L ∞ (Ω) topology. This result is essentially classical, at least for bounded domains. We give a proof at the end of the section.

Note that, in general, this result fails for L ∞ loc topology. For example, consider the Allen-Cahn equation in R, -u = u(1 -u 2 ). This equation admits an explicit solution u : x → tanh x √ 2 which is stable (degenerate). On the one hand, the family of the translated solutions u a (•) = u(• -a) converges to 0 when a → +∞ in the L ∞ loc topology. On the other hand, 0 is a stable non-degenerate solution (because f (0) < 0). One could argue that the above counterexample relies on the fact that the nonlinearity is balanced, that is, 1 0 f = 0 with f (u) := u(1 -u)(u -1 /2). However, one can build similar counterexamples for unbalanced nonlinearities by considering a ground state (which has been proved to exist in most cases, see, e.g., [START_REF] Berestycki | Nonlinear scalar field equations, I. Existence of a ground state[END_REF]).

In an attempt to extend the results of section 4, we address the following question. Question 3 Is Σ u from (20) always a singleton when u is stable non-degenerate?

We think the answer is negative; yet, we are not able to provide a counterexample.

Proof (Lemma C.1) Assume that there exists u k ∈ S, u k+1 ≡ u k , a sequence which converges to u, and let us show that λ 1 = 0. We set v k := u k+1 -u k . For all k, u k is a solution of (1) in Ω, thus

-∆v k -c k (x)v k = 0 in Ω, ∂ ν v k = 0 on ∂Ω, ( 30
)
where c k (x) := f (u k+1 (x)) -f (u k (x)) u k+1 (x) -u k (x) .

Since f is C 1,α and u k+1 -u k is bounded, c k (x) converges uniformly to f (u(x)) when k → +∞. Formally, we have

F v k v k L 2 ≤ f (u) -c k ∞ -→ k→0 0,
(with F from (2)) which contradicts the fact that u is stable non-degenerate. However, the former calculation is not licit when Ω is unbounded. To make it rigorous, we use the cut-off function χ R defined in [START_REF] Berestycki | Nonlinear scalar field equations, I. Existence of a ground state[END_REF]. Multiplying [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF] by v k χ 2 R , integrating on Ω, using the divergence theorem and the boundary condition in [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF] we find

F χ R v k χ R v k L 2 = Ω χ 2 R v 2 k (c k -f (u)) Ω χ 2 R v 2 k + Ω |∇χ R | 2 v 2 k Ω χ 2 R v 2 k ≤ c k -f (u) L ∞ (Ω R ) + 4 α k,R ,
where

C k (R) := Ω R v 2 k , α k,R := C k (2R) R 2 C k (R) .
On the one hand, in the proof of Theorem 1.4, we show that, for fixed k ≥ 0, lim inf R→+∞ α k,R ≤ 0. On the other hand, since c k -f (u) L ∞ (Ω R ) goes to 0 when k → +∞, uniformly in R, we deduce that F can be made arbitrarily small. It implies λ 1 ≤ 0. The reverse inequality λ 1 ≥ 0 can be proved with the same method, which achieves the proof.

Figure 1 :

 1 Figure 1: Examples of asymptotically convex cylinder

  a principal eigenfunction associated with λ 1 (u, Ω) (given by Proposition A.2), define a n := ϕ[y n ](0) and ϕ n := 1 an ϕ[y n ]. Note that λ 1 (u[y n ], Ω[y n ]) = λ 1 (u, Ω). We have

Proposition 4 . 9

 49 Assume Ω is asymptotically θ-invariant (Definition 3.4) and satisfies sup (x,r,θ)∈Ω r < +∞.

  and consider the eigenvalue problem with mixed boundary conditions -Lψ = λψ a.e. in Ω R , Bψ = 0 a.e. on ∂Ω ∩ D R , ψ = 0 a.e on Ω ∩ ∂D R .

  2,α function (still denoted u) which is defined in R n , coincides with u on Ω, and is identically 0 in R n \U , U being a neighborhood of Ω. Proof As u satisfies Neumann boundary conditions, we can extend it by reflexion on an open set U ⊃ Ω. Note that, since the domain is uniformly C 2,α , we can choose U such that

	inf
	(x,y)∈Ω×∂U

If the variant of De Giorgi conjecture were to be true in dimension n = 8, the method of Ambrosio and Cabré[START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in Rˆ3 and a conjecture of De Giorgi[END_REF] would imply that De Giorgi conjecture holds in dimension n = 9 which is impossible from the result of del Pino, Kowalczyk and Wei[START_REF] Del Pino | A counterexample to a conjecture by De Giorgi in large dimensions[END_REF]. See also[START_REF] Pacard | Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones[END_REF].

The author is deeply thankful to Professor Henri Berestycki for proposing the subject and for all the very instructive discussions. The author also thanks the anonymous referee for his comments which helped to improve the paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi -ReactionDiffusion Equations, Propagation and Modelling held by Henri Berestycki.

Proposition 3. [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] Let Ω be a θ-invariant domain which satisfies [START_REF] Caffarelli | Phase transitions: Uniform regularity of the intermediate layers[END_REF] and u be a stable solution of [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF]. Assume that [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF] holds or that u is stable non-degenerate. Then ∂ θ u = 0.

Proof We set v := ∂ θ u. Note that, in a Cartesian system of coordinates

We also know from [START_REF] Caffarelli | Phase transitions: Uniform regularity of the intermediate layers[END_REF] that v is bounded. Differentiating (1), we find that v satisfies [START_REF] Berestycki | One-dimensional symmetry of bounded entire solutions of some elliptic equations[END_REF]. Moreover, as Ω is θ-invariant, we have ∂ ν ∂ θ u = 0 on ∂Ω. We can therefore proceed as in the proof of Theorem 1.4 to prove that if v ≡ 0 then λ 1 ≤ 0. It completes the proof for the case where u is stable non-degenerate.

Assume now that (5) holds. Since v satisfies ( 14) and ( 15), Lemma 3.

Therefore v ≡ 0, which completes the proof. [START_REF] Matano | Asymptotic behavior and stability of solutions of semilinear diffusion equations[END_REF], since ∂ θ u = 0, then w := ∂ r u satisfies

Remark 3.6 As noticed by Matano

and w satisfies [START_REF] Bombieri | Minimal cones and the Bernstein Problem[END_REF]. Therefore, if we further assume that Ω is convex, then Theorem 1.4 and Theorem 1.5 apply. It proves the non-existence of patterns in some non-convex domains, such as rings or torus. Similar strategies and refined results can be found in the work of Alikakos and Bates [START_REF] Alikakos | On the singular limit in a phase field model of phase transitions[END_REF]. In this work, the authors investigate in particular the existence of patterns when the equation features a radial source term, the domain is a ball or an annulus, and the reaction term is multiplied by a large factor ε -1 with ε 1.

Asymptotic symmetries

In this section, we establish asymptotic symmetries for patterns. We prove Theorem 1.6 and also the following analogous result for possibly degenerate patterns.

Theorem 4.1 Under the same assumptions as in Theorem 1.6, but with u a bounded pattern which is possibly degenerate. Suppose that the stable roots of f , denoted (z i ), are isolated, and that zj zi f = 0, for i = j. Let us further assume that the limiting domain Ω ∞ satisfies [START_REF] Bandle | Existence and nonexistence of patterns on Riemannian manifolds[END_REF]. Then, u(x 1 , •) converges C 2 loc to a stable root of f when x 1 → +∞.

In section 4.2, we also adapt the symmetry results of 4.2 when the domain is asymptotically invariant under a translation or a rotation.

First, let us give a precise definition of the convergence of a domain.

Definition 4.2

Let Ω ⊂ R n be a uniformly smooth domain. For any y ∈ R, we define the translated domains (on the x 1 -axis)

We say that Ω converges to Ω ∞ ⊂ R n if the boundary of Ω[y] converges to that of Ω ∞ when y → +∞ in the C 2,α loc topology.

Convergence to a constant -proof of Theorem 1.6 and Theorem 4.1

Assume Ω ⊂ R n is uniformly smooth and converges to a convex domain Ω ∞ ⊂ R n . Let u be a solution of [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF]. For technical reasons, we need to extend u in all R n .


E Acurio
email: e.acurio@dimes.unical.it

F Crupi

P Magnone

L Trojman

G Meneghesso

F Iucolano

On Recoverable Behavior of PBTI in AlGaN/GaN MOS-HEMT

Keywords: PBTI, AlGaN/GaN MOS-HEMT, oxide traps, recessed gate, SiO2

This experimental study focuses on the positive bias temperature instability (PBTI) in a fully recessed-gate AlGaN/GaN MOS-HEMT. A positive stress voltage to the gate results in positive threshold voltage shift (ΔVth), which is attributed to the trapping of electrons from the GaN layer into the pre-existing oxide traps. The trapping rate exhibits a universal decreasing behavior as a function of the number of filled traps, independently of stress time, stress voltage, stress temperature, and device-to-device variability. The stress-induced ΔVth can be fully recovered by applying a small negative voltage, which causes the electron de-trapping. In the explored time window (between 1 s and thousands of s), the recovery dynamics is well described by the superimposition of two exponential functions associated with two different traps. Both trap time constants are independent of the stress voltage, decrease with temperature and increase with the recovery voltage. The activation energy of the slower trap is 0.93 eV, while the faster trap exhibits an activation energy with a large spread in the range between 0.45 eV and 0.82 eV.

Introduction

In the field of power electronics, there is a continuous requirement of advanced power transistors, enabling to improve the efficiency and the power density of converters. AlGaN/GaN HEMTs have been recently considered as a possible replacement to conventional Si-based devices [START_REF] Baltynov | The world's first high voltage GaN-on-Diamond power semiconductor devices[END_REF]- [START_REF] Saito | Recessed-Gate Structure Approach Toward Normally Electronics Applications[END_REF]. Indeed, GaN-based devices benefit of an intrinsic better tradeoff between onresistance and breakdown voltage, enabling also the operation at very high frequencies [START_REF] Schwierz | RF transistors: Recent developments and roadmap toward terahertz applications[END_REF]- [START_REF] Saadat | Gate-first AlGaN/GaN HEMT technology for high-frequency applications[END_REF].

The potentiality of GaN HEMT has been largely proved in literature, and although the reliability of such a device is still the main concern [START_REF] Meneghesso | Reliability of GaN high-electron-mobility transistors: State of the art and perspectives[END_REF][10], commercial power devices are already available. Nevertheless, an important limitation of GaN HEMTs is their normally-on behavior. Although cascade solutions can be implemented, there is a significant effort in developing normally-off solutions [START_REF] Uemoto | Gate injection transistor (GIT) -A normally-off AlGaN/GaN power transistor using conductivity modulation[END_REF] [START_REF] Kim | Normally-off GaN MOSFETs on insulating substrate[END_REF]. The device considered in this paper is an AlGaN/GaN MOS-HEMT, in which a metal/oxide/ semiconductor stack is adopted to switch-off the current at zero gate voltage. The performance of AlGaN/GaN HEMTs is typically limited by the current collapse phenomenon, which consists of a current reduction due to the application of a negative stress [START_REF] Jin | Methodology for the study of dynamic ON-resistance in high-voltage GaN field-effect transistors[END_REF]- [START_REF] Arulkumaran | Study of current collapse by quiescent-bias-stresses in rf-plasma assisted MBE grown AlGaN/GaN high-electron-mobility transistors[END_REF]. In the case of MOS structure, another important issue affecting device performance is the bias temperature instability (BTI), i.e. a degradation of electrical parameters (such as threshold voltage and charge carrier mobility) induced by the application of high gate voltages and high temperatures. This phenomenon has been extensively studied in CMOS devices for logic applications [START_REF] Crupi | Positive bias temperature instability in nMOSFETs with ultra-thin Hf-silicate gate dielectrics[END_REF]- [START_REF] Franco | On the recoverable and permanent components of Hot Carrier and NBTI in Si pMOSFETs and their implications in Si0.45Ge0.55 pMOSFETs[END_REF]. Recently, experimental studies on BTI have been performed also in the case of AlGaN/GaN MOS-or MIS-HEMTs [START_REF] Crupi | Low Frequency Noise and Gate Bias Instability in Normally OFF AlGaN/GaN HEMTs[END_REF]- [START_REF] Wu | Toward Understanding Positive Bias Temperature Instability in Fully Recessed-Gate GaN MISFETs[END_REF], in order to analyze the impact of different gate dielectrics and to understand the physics associated with the defects responsible of BTI (spatial and energy distribution).

In [START_REF] Crupi | Low Frequency Noise and Gate Bias Instability in Normally OFF AlGaN/GaN HEMTs[END_REF] we have recently conducted a correlation study between 1/f noise and BTI degradation in AlGaN/GaN MOS-HEMTs. We found out that defects originating 1/f noise are responsible for a significant dispersion in the fresh threshold voltage, while different defects (perhaps with different energy or spatial localization) lead to BTI effect. Following the study in [START_REF] Crupi | Low Frequency Noise and Gate Bias Instability in Normally OFF AlGaN/GaN HEMTs[END_REF], this paper aims at carrying out a more comprehensive analysis of BTI on such devices, by evaluating and modeling the influence of stress voltage and temperature on the threshold voltage shift. Moreover, we investigate the time constants and the activation energy of traps involved during BTI. Even though AC measurement techniques have been proposed to capture the behavior of fast traps [START_REF] Lansbergen | Threshold voltage drift (PBTI) in GaN D-MODE MISHEMTs: Characterization of fast trapping components[END_REF], we focused our work on the study of the slow components of the threshold voltage degradation by using DC characterization.

The remainder of the paper is organized as follows: in Section 2 we provide the details of the devices under test and of the characterization methodology; in Section 3 we report and discuss the results of the experimental analysis; finally in Section 4 the main achievements of the paper are summarized.

Experimental

The structure of the AlGaN/GaN MOS-HEMT studied in this work is shown in Fig. 1. It is grown Al0.25Ga0.75N/GaN heterostructure on 150 mm Silicon (111) substrates in a MOCVD reactor. First of all, the ohmic contacts were formed evaporating the Ti/Al-based metallizations and carrying out a rapid thermal annealing at 850 °C [START_REF] Iucolano | Correlation between microstructure and temperature dependent electrical behavior of annealed Ti/Al/Ni/Au Ohmic contacts to AlGaN/GaN heterostructures[END_REF]. After that, the recession of the AlGaN layer was performed with a dry chemical process based on chlorine to ensure a normally-off operation. The 50 nmthick SiO2 layer was deposited by plasma enhanced chemical vapor deposition (PECVD) by using a tetraethyl orthosilicate precursor, followed by a thermal annealing at 850 °C in N2 [START_REF] Crupi | Low Frequency Noise and Gate Bias Instability in Normally OFF AlGaN/GaN HEMTs[END_REF]. Gate contacts were evaporated on the SiO2 by using Ni-based metals [START_REF] Iucolano | Influence of properties of Si3N4 passivation layer on the electrical characteristics of Normally-off AlGaN/GaN HEMT[END_REF].

In order to study the complete dynamics of the trapping and de-trapping process in our devices, a set of stress-recovery experiments was performed at different stress voltages and temperatures with the parameter analyzer Keithley 4200-SCS. All measurements were done at high temperatures ranging from 75°C to 190°C and with VDS=50 mV. The threshold voltage is determined using the fixed current criterion at 10 µA. The adopted measurement procedure consists of three phases: initial stabilization, stress, and recovery.

The initial stabilization is performed by applying a negative gate voltage (typically -1 V) for 3000 s. During this phase, the virgin device is stabilized by releasing the charges originally contained in trapping centers [START_REF] Wu | Analysis of slow de-trapping phenomena after a positive gate bias on AlGaN/GaN MIS-HEMTs with in-situ Si3N4/Al2O3 bilayer gate dielectrics[END_REF] [START_REF] Zhang | Observation of threshold voltage instabilities in AlGaN/GaN MIS HEMTs[END_REF]. It is worth noting that without applying the initial stabilization, positive and negative ΔVth were observed in different samples and in different stress conditions, due to the concomitant charge trapping and releasing during the stress. After this phase, we measured a complete ID-VGS curve sweeping VGS from -3 V to 3 V. It was necessary to measure the curve up to 3 V, which is significantly higher than Vth in fresh device, in order to be able to measure the Vth after stress also in the case of high ΔVth shifts. As shown in Fig. 2, the ID-VGS curve after the initial stabilization is significantly different from the ID-VGS curve measured in a virgin device. This stabilization is performed each time that we change the temperature because it allows the device to reach a reproducible reference state for the subsequent experiments. As a matter of fact, after the recovery stage following the stress phase for different stress voltages, the ID-VGS curve always comes back to this reference state.

During the stress phase, the device is biased with a positive gate voltage (Vstress) ranging between 6 V and 14 V.

Temperatures above 190°C and higher voltages were not used in order to avoid oxide breakdown. In order to monitor the evolution of the stress-induced degradation, the stress was interrupted at fixed time intervals and an ID-VGS curve was measured and compared with the reference curve obtained in the stabilization step to calculate ΔVth.

Immediately after the stress phase, we executed the recovery phase, by biasing the device with a gate voltage (Vrecovery) ranging between 0 V and -2 V for at least 1000 s. Also in this phase, in order to monitor the recovery evolution, we interrupted the recovery at fixed time intervals and an ID-VGS curve was measured and compared with the reference curve obtained in the stabilization step. Fig. 2 illustrates that by applying the recovery procedure after the stress, the ID-VGS curve comes back to the initial stabilized state indicating that no permanent damage was introduced during the stress phase. It is worth noting that the absence of permanent damage is related to the investigated gate voltage values during the stress (6 V to 14 V) and it cannot be extended to higher gate voltages.

Results and discussion

A. Stress Phase

In Fig. 3 we report the evolution of ID-VGS curves with stress in the case of a representative sample. It can be seen that after each stress interval, the ID-VGS curve moves toward the positive VGS direction, showing the notorious threshold voltage degradation. The observed ΔVth shift can be ascribed to the electron trapping from the channel (under the gate oxide) into the traps located in the SiO2 energy gap (see Fig. 4 (a)). Fig. 5 shows ΔVth during the stress at different stress voltages and temperatures. In order to determine the ΔVth evolution, we evaluated the trapping rate parameter defined as . It can be noted that if b is constant, ΔVth evolution follows a classic power law function, i.e.

[31] [START_REF] Mahapatra | A predictive reliability model for PMOS bias temperature degradation[END_REF]. As shown in Fig. 6 (a), the trapping rate parameter significantly decreases during the stress, thus indicating that ΔVth evolution does not follow a power law. Fig. 6 (b) and Fig. 6 (c) also illustrates that b considerably declines by increasing stress voltage and temperature.

In order to understand the physics behind these phenomena, we plot the trapping rate parameter b, which is a measure of the PBTI charging rate, as a function of the number of trapped charges per unit area, ΔNOT (see Fig. 7), for different experimental conditions (i.e. time, voltage and temperature) and different samples. The density of trapped charges has been estimated as ΔNOT = (COX *ΔVth)/q, where COX and q are the oxide capacitance per unit area and the elementary charge, respectively. The overall measurement results illustrate a universal decreasing behavior of the charging rate as a function of the number of filled traps independent of stress time, stress voltage, stress temperature, and device-to-device variability. This trend suggests that the probability of charging traps is associated with the number of available empty traps that exist in the analyzed time window. As an additional cause, this decreasing tendency could be explained by a charge feedback mechanism in which the electrons accumulated near or in the SiO2/GaN interface during the stress increase the barrier potential and diminishes the effect of the forward bias [33][34]. The longer the stress, the more the barrier potential is increased and this could be related to the decreasing trend of the trapping parameter b observed in Fig. 6 (a).

Additionally, we observe a linear correlation by plotting the transconductance variation as a function of the threshold voltage shift for different stress voltages in Fig. 8. This observation suggests that there is also the additional contribution of the charge trapping at the SiO2/GaN interface [START_REF] Zhang | Observation of threshold voltage instabilities in AlGaN/GaN MIS HEMTs[END_REF][24], which causes mobility degradation.

B. Recovery Phase

Fig. 9 shows the relaxation data measured after different stress voltages, temperatures and recovery voltages. For a sufficiently high recovery time, ΔVth tends to zero, thus indicating that in our experimental conditions no permanent damage was introduced during the stress phase. In other words, the observed PBTI is fully recoverable. The observed ΔVth evolution during the recovery phase is ascribed to the release of electrons from the energy states of the SiO2 band-gap into the GaN layer (see Fig. 4 (b)). Several models have been proposed to describe the BTI relaxation dynamics [START_REF] Joh | A Current-Transient Methodology for Trap Analysis for GaN High Electron Mobility Transistors[END_REF]- [START_REF] Pobegen | On the Distribution of NBTI Time Constants on a Long, Temperature-Accelerated Time Scale[END_REF]. As shown in Fig. 9, we considered a simple model based on the superimposition of two exponential functions "

where A1 and A2 represent the amplitudes and τ1 and τ2 are the time constants. The good agreement between experimental data and fitting lines indicates that in the observed time window (between 1s and thousands of seconds), the recovery dynamics under different conditions is well described by two effective sets of oxide traps located at different energy and/or position [START_REF] Tallarico | Modeling spatial and energy oxide trap distribution responsible for NBTI in p-channel power U-MOSFETs[END_REF]. As an

t V b th log log ∂ Δ ∂ = b stress th at V = Δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = Δ 2 2 1 1 exp exp τ τ t A t A V th
example, we can observe the final fitting and the individual contribution of trap 1 and 2 for the recovery data after Vstress=14V in Fig. 10. As shown in Fig. 11 (a), both time constants are independent of the stress voltage (between 6V and 14V). This result suggests that the same sets of traps have been filled independently of the applied stress voltage. From the Fig. 11 (b) it can be seen that the two-time constants strongly decrease with temperature. Furthermore, both time constants decrease with more negative gate voltage (Fig. 11 (c)). As previous studies indicate [START_REF] Arulkumaran | Study of current collapse by quiescent-bias-stresses in rf-plasma assisted MBE grown AlGaN/GaN high-electron-mobility transistors[END_REF][17], a reverse electric field accelerates the de-trapping process of the charges by enhancing the electron tunneling back to the semiconductor layer. The main limitation of using a negative gate voltage for the recovery is to find the appropriate value to rapidly release charges from the traps without inducing a new stress phase.

In addition, we illustrate the ratio between the two amplitudes which is independent of the stress voltage (Fig. 11 (d)) and gradually increases with temperature (Fig. 11 (e)) and recovery voltage (Fig. 11 (f)). It is worth noting that the ratio less than one implies a dominant trap characterized by time constant τ2.

In general, a wide distribution of time constant is expected for such a device, since a Gaussian distribution of the energy barrier is foreseen for the emission of single defects. Therefore, we tried to fit the relaxation data with a higher number of time constants. By comparing these results with the ones obtained by using only two exponentials, we observe that the data fitting is only slightly improved, but the trends of the time constants are quite noisy and unstable due to the limited set of experimental points. For this reason, we prefer to adopt the parsimonious model with only two effective time constants, but this is a simplified and purely empirical description of the underlying complex physical mechanism behind the BTI relaxation phase. Furthermore, we evaluated the fitting of the recovery data by using the universal relaxation law [START_REF] Grasser | The Universality of NBTI Relaxation and its Implications for Modeling and Characterization[END_REF] with the assumption of a negligible permanent component. The trends of the β parameter (see Eq. 8 in [START_REF] Grasser | The Universality of NBTI Relaxation and its Implications for Modeling and Characterization[END_REF]) as a function of experimental parameters (stress voltage, stress temperature and recovery voltage) completely agree with the results obtained by using the sum of two exponentials (clearly, since β measures the recovery speed, the observed agreement means that β shows the opposite behavior of the two-time constants). However, we prefer to adopt the two-time constants model since it shows a better match with the experimental data.

In order to determine the apparent activation energies for the two sets of traps, we used the Arrhenius plot reported in Fig. 12. The two extracted activation energies, Ea1 and Ea2, in diverse samples and by applying different recovery voltages can be found in Fig. 13. On the one hand, the average activation energy of the slower trap (Ea2) is 0.93 eV and this value is almost independent of the recovery voltage and the investigated sample. On the other hand, the estimated value of the activation energy of the faster trap (Ea1) exhibits a significantly higher dispersion, falling in the range between 0.45 eV and 0.82 eV. The larger energy spread is ascribed to the lower accuracy in the evaluation of the time constant of the faster trap, due to the lower amplitude. It is important to highlight that the extracted activation energies are related to the traps that exist in the analyzed time window (1 to 100 s) and cannot be extended to fast traps in the regime of µs and ns.

Conclusions

The observed completely recoverable PBTI in AlGaN/GaN MOS-HEMT is ascribed to the initial trapping of electrons from the GaN layer into the traps located in the SiO2 energy gap (stress phase) and the successive electron de-trapping (recovery phase). The PBTI trapping rate parameter exhibits a universal decreasing behavior during the stress phase as a function of the number of filled traps, independently of stress time, stress voltage, stress temperature, and device-to-device variability. In the observed time window (between 1 s and thousands of s), the recovery dynamics is well described by the superimposition of two exponential functions. Both time constants decrease for more negative recovery voltages and for higher temperatures, while they are stress-voltage independent. Relaxation measurements at different temperatures indicate that the slower trap, which is the dominant, exhibits an average activation energy of 0.93 eV, while the average activation energy of the faster trap shows a large spread in the interval between 0.45 eV and 0.82 eV. ΔN OT (cm -2 )

75°C (A) 100°C (A) 125°C (A) 150°C (A) 75°C (B) 100°C (B) 125°C (B) 150°C (B) 6V (A) 8V (A) 10V (A) 14V (A) 6V (B) 8V (B) 10V (B) 14V (B) 150°C_14V (

  Figures:

Figure 1 :Figure 2 :Figure 4 :"Figure 5 :Figure 7 :

 12457 Figure 1: Schematic of AlGaN/GaN MOS-HEMT analyzed in this work."

Figure 6 :

 6 Figure 6: The trapping rate parameter " significantly decreases during the stress (a), and by increasing stress voltage (a) and temperature (b).

Figure 8 :Figure 9 :Figure 10 :"Figure 11 :

 891011 Figure 8: Transconductance variation vs. correlation is observed (straight line).

Figure 12 :

 12 Figure 12: Arrhenius plot of the characteristic recovery time constants extracted from the exponential fitting during the recovery phase. The two activation energies are obtained by the slope of the fitting lines.

Figure 13 :

 13 Figure 13: Apparent activation energy of the two traps (Ea1 and Ea2) obtained in different samples (A, B, C and D) at different recovery voltages. These values include the acceleration of trapping and de-trapping at the same time while the chuck is maintained at the same temperature during the whole measurement. The activation energy of the faster trap (Ea1) exhibits a lower mean value and a higher dispersion.

Acknowledgment

The research leading to these results was supported by the ENIAC-324280 Energy Efficient Converters using GaN Power Devices (E2COGaN).