On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in ${\bf R}^2$

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension n=2 an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.


Introduction
We consider the Allen-Cahn equation where Ω ⊂ R n is a bounded or unbounded domain, g : ∂Ω → R is continuous and bounded and W : R → R is a C 3 potential.
We are interested in symmetric solutions: where for z ∈ R d we letẑ = (−z 1 , z 2 , . . . , z d ) the reflection in the plane z 1 = 0. We assume: h 1 − W : R → R is an even function: W (−u) = W (u), for u ∈ R, (1.2) which has a unique non-degenerate positive minimizer: 3 − Ω ⊂ R n is a domain with nonempty boundary which is symmetric: x ∈ Ω ⇒x ∈ Ω, (1.5) and of class C 2+α . If Ω is unbounded we require that Ω satisfies a uniform interior sphere condition and that the curvature of ∂Ω is bounded in the C α sense.
If S ⊂ R d is a symmetric set, we define S + := {x ∈ S : x 1 > 0}. We first consider the case of general n ≥ 1 and prove the existence of a symmetric solution which is near 1 in Ω + . Note that, in general, ∂(Ω + ) = (∂Ω) + . Theorem 1.1. Assume that W and Ω ⊂ R n satisfy h 1 , h 2 and h 3 . Assume that g : ∂Ω → R is symmetric and bounded as a C 2,α (∂Ω; R) function and satisfies g(x) ≥ 0, for x ∈ (∂Ω) + .
(We assume that g is extended to Ω as a symmetric C 2,α map). A similar statement is valid in the case of Neumann boundary conditions. We then restrict to the case n = 2 and prove the following asymptotic result The map q depends only on W, n and on the C 1 (Ω; R) norm of u.
A convergence result for odd solutions of (1.1) similar to (1.8) valid in the case Ω ⊂ R n is a half space was obtained, among other things, in [2] (cfr. Theorem 1.1). The point in Theorem 1.2 is that, even though is restricted to n = 2, applies to general domains that satisfy (1.7). Some of the ideas in the proof of Theorem 1.2 have been extended and utilized in [1] where the restriction to n = 2 is removed and u is allowed to be a vector.
The proof of Theorems 1.1 and Theorem 1.2 is variational and is based on a pointwise estimate for local minimizers of the Allen-Cahn energy defined for all bounded domain A ⊂ R n and u ∈ W 1,2 (A; R).

Definition.
Let Ω ⊂ R n be a domain. A map u ∈ W 1,2 loc (Ω; R) is a local minimizer of the Allen-Cahn energy if for every bounded Lipschitz domain A ⊂ Ω.
In the following we denote by k, K and C generic positive constants that can change from line to line.
The pointwise estimate alluded to above is stated in the following Let Ω ⊂ R n a domain and u ∈ C 1 (Ω; R) a local minimizer of the Allen-Cahn energy that satisfies Then there is q * > 0 with the property that for each q ∈ (0, q * ] there is R(q) > 0 such that Moreover R(q) can be chosen strictly decreasing and continuos in (0, q * ]. The inverse map q(R) satisfies q(R) ≤ Ke −kR , R ∈ [R(q * ), +∞), (1.14) for some positive constants k, K that depend only on W, n and the bound M 0 .
The paper is organized as follows. In Sect. 2 we use Theorem 1.3 to prove Theorem 1.1. In Sect. 3 we prove Theorem 1.2. Finally in Sect. 4 we present a proof of Theorem 1.3. The proof is an adaptation to the scalar case of arguments developed in [3] and [4] for the vector Allen-Cahn equation.
2 The proof of Theorem 1.1 We first consider the case of Ω bounded. Then standard arguments from variational calculus yield the existence of a symmetric minimizer u ∈ g + W 1,2 0,S (Ω; R) of J Ω ; we denote by W 1,2 0,S (Ω; R) the subspace of symmetric maps of W 1,2 0 (Ω; R). Let g m = max (∂Ω) + g. We can assume |u| ≤ M ′ := max{M, g m }, (2.1) and and observe that if Ω + ∩ {u > M ′ } has positive measure, then h 2 implies (2.4) in contradiction with the minimality of u. The proof of (2.2) is similar.
From the bound (2.1), the smoothness assumption on ∂Ω in h 3 and elliptic regularity we obtain that u is a classical solution of (1.1) and for some constant M ′′ > 0. The restriction of u to Ω + trivially satisfies the definition of minimizer of the Allen-Cahn energy in Ω + with potentialW that, by (2.1) and (2.2), can be identified with any smooth function that satisfiesW (s) = W (s), for s ≥ 0 and W (s) > W (|s|), for s < 0. From this and (2.5) it follows that we can apply Theorem 1.3 toû = u − 1 with potentialW (· + 1) and conclude that u satisfies the exponential estimate with k, K depending only on W and M ′′ . This concludes the proof for Ω bounded. If Ω is unbounded we consider a sequence of bounded domains Ω j , j ∈ IN, such that Ω j ⊂ Ω j+1 and Ω = ∪ j Ω j . From h 3 we can assume that the boundary of Ω j is of class C 2,α and satisfies an interior sphere condition uniformly in j ∈ IN. Therefore the same reasoning developed for the case of bounded Ω yields and The estimate (2.7) implies that, passing to a subsequence if necessary, we can assume that u j converges locally in C 2 to a classical solution u : Ω → R of (1.1) and (2.8) implies that u satisfies the exponential estimate in Theorem 1.1. The proof of Theorem 1.1 is complete.
Remark. Elliptic regularity implies that we can upgrade the exponential estimate in Theorem 1.1 to In the proof of Theorem 1.2 below we make systematic use of the fact that the solution of (1.1) given by Theorem 1.1 is a local minimizer in the sense of Definition 1. This is obvious when Ω is a bounded. If Ω is unbounded it follows from the fact that u = lim j→+∞ u j is the limit of a sequence of minimizers u j : Ω j → R, Ω j bounded, that converges to u uniformly in compacts [7].

The proof of Theorem 1.2
We divide the proof in several lemmas.
For w ∈ W 1,2 ((−l, l); R) we set Lemma 3.1. There exist q 0 > 0, c > 0, independent of l > 1, such that for v ∈ B l , v = qν, we have Moreover it results Proof. From (1.9) and v(±l) = 0 it follows where we have also added and subtracted 1 2 W ′′ (u)v 2 . By differentiating (1.9) we see thatū x 1 is an eigenfunction corresponding to the eigenvalue λ = 0 for the operator L defined by Sinceū is increasing and odd,ū x 1 is positive and even. On the other hand the assumption W ′′ (±1) > 0 implies, see e.g. Theorem A.2 of [6] (pag. 140), that the essential spectrum of L is contained in [a, +∞) for some a > 0. Therefore, if we restrict to the subset of odd functions we can conclude that there exists a positive constant c 1 such that In particular, given v ∈ B l , we can apply (3.10) to the trivial extensionṽ of v to obtain Since Then, for some map From (3.9), (3.11) and (3.13), if we choose q 0 > 0 so small that CW ′′′ q To show (3.6) let us consider the minimization problem It is easy to construct a smooth odd map w ∈ B l that satisfies the constraint w l ≥ q 0 . Therefore there exists a minimizing sequence {v j } ⊂ B l that satisfies v j l ≥ q 0 , j ∈ IN, and From (3.15) and standard arguments from variational calculus it follows that there is v l ∈ B l and a subsequence {v j h } such that It follows v l l ≥ q 0 and v l is a minimizer of (3.14). Since E l (0) = 0 and v = 0 is the unique minimizer of E l on B l , this implies E l (v l ) = α l > 0, and therefore Note that α l is non increasing with l. Indeed, if l 1 < l 2 and v ∈ B l 1 , then the trivial extensionṽ of v to [−l 2 , l 2 ] satisfies E l 2 (ṽ) = E l 1 (v) and belongs to B l 2 . Therefore, there exists lim l→+∞ α l . We claim that Let {l k } k be a sequence of positive numbers such that l k → +∞ for k → +∞. Let v k be a minimizer of problem (3.14) for l = l k and letṽ k : R → R be the trivial extension of v k , we may assume that the sequence {ṽ k } k converges in L 2 (R) and weakly in W 1,2 (R) to a map v which satisfies, by lower semicontinuity of for a suitable constant c 2 independent of l. Then both (3.5) and (3.6) hold with c := min{c 1 , c 2 }. To prove (3.7) we note that setting v = qν in (3.9) yields This concludes the proof.
For r > 0, l > 0 and η ∈ R, we denote by C r l (η) the set In the following, whenever possible, we assume that by a translation we can reduce to the case η = 0 and write simply C r l instead of C r l (0). The introduction of the map E l allows to represent the energy J C r l (v) of an odd map v : C r l → R that satisfies v(x) =ū(x 1 ), for x 1 = ±l, |x 2 | < r in a particular form that we now derive. We have For later reference we state Then, there is a constant C 2 > 0 independent of l ≥ 1 such that , for x ∈ C l,r and x ∈ Ω \ C r+δ l+δ ; where δ > 0 is a fixed constant.
Arguments analogous to the ones in the proof of Lemma 3.3 prove Lemma 3.4. Assume that C r l satisfies (3.27). Then there is a Lipschitz continuous function v with the following properties: for some constants C, γ > 0. Lemma 3.5. Let q 0 and c be as in Lemma 3.1. Given q < q 0 , fix r > 0 such that where J 0 is the constant in (iv) in Lemma 3.3. There is l(q) > 0 such that, provided (3.27) is satisfied with l ≥ max{l r , l(q)}, then there exist a − ∈ (−r, −r/2) and a + ∈ (r/2, r) such that u(·, a ± ) −ū l+δ/2 <q.
To prove (5.3) note that, for q ∈ (0, q * ), Analogously, for q ∈ (−q * , 0), By reducing the value of q * if necessary, we can also assume where W > 0 is a suitable constant. This follows from assumption (iii) and (1.12). All the arguments that follow have a local character. Therefore, without loss of generality, in the remaining part of the proof we can assume that Ω is bounded.
Proof. Let b > 0 be a number such that b ≤ min x∈B x 0 ,R ϕ. Since u is continuous the set A b := {x ∈ B x 0 ,R : u > b} is open and there exists a function ρ + ∈ W 1,2 (A b ) that minimizes the functional J A b (p) = A b ( 1 2 |∇p| 2 + W (p))dx in the class of functions that satisfy the Dirichlet condition p = u on ∂A b . Since |ρ + |+ρ + 2 is also a minimizer we have ρ + ≥ 0. We also have ρ + ≤q. This follows from (5.3) and (5.4) which imply that min{ρ + ,q} is also a minimizer. The map ρ + satisfies the variational equation If we take η = (ρ + − ϕ) + in (5.10) and use (5.3) we get
Let v 1 j , v 2 j ∈ W 1,2 (Ω) be the maps defined as follows: v 1 j is the map v defined in Lemma 5.4 for B x 0 ,R+λ with R = R + jλ. v 2 j is the map v given by Corollary 5.3 when u = v 1 j and R = R + jλ. From these definitions, Corollary 5.3 and Lemma 5.4, we deduce By adding these inequalities and using the minimality of u we obtain 0 ≥ J Ω (u) − J Ω (v 2 j ) ≥ kσ j − K(σ j+1 − σ j ) (5.43) and therefore, where ω is the measure of the unit ball in R n . For j sufficiently large the last inequality is not satisfied and this contradicts the minimality of u. We denote j m the minimum value of j such that (5.44) is violated. Then, (5.39) follows with R 0 = R + (j m + 1)λ.
The existence of the map (0, q * ] ∋ q → R(q) follows from the fact that all the above arguments can be repeated with a generic q ∈ (0, q * ) in place of q * . We can obviously assume that R(q) is decreasing and, by modifying it if necessary, we can also assume that it is strictly decreasing and continuous.
For completing the proof of Theorem 1.3 it remains to prove the estimate (1.14). Proposition 5.5 and in particular (5.39) imply that we can apply Lemma 5.2 to u and the ball B x,R for each x ∈ Ω such that d(x, ∂Ω) = R 0 + R with R ≥ R 0 . Therefore we obtain |u(x)| ≤ φ(0, R). We also have (see [5]) that φ(0, R) ≤ q * e −k 0 R = q * e k 0 R 0 e −k 0 d(x,∂Ω) ,