Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

Abstract

Abstract. We study the existence of solutions u : R 3 → R 2 for the semilinear elliptic systems − ∆u(x, y, z) + ∇W (u(x, y, z)) = 0, (0.1) where W : R 2 → R is a double well symmetric potential. We use variational methods to show, under generic non degenerate properties of the set of one dimensional heteroclinic connections between the two minima a ± of W , that (0.1) has infinitely many geometrically distinct solutions u ∈ C 2 (R 3 , R 2 ) which satisfy u(x, y, z) → a ± as x → ±∞ uniformly with respect to (y, z) ∈ R 2 and which exhibit dihedral symmetries with respect to the variables y and z. We also characterize the asymptotic behaviour of these solutions as |(y, z)| → +∞.

Finite Morse Index Implies Finite Ends

“…Near each end the solution approaches to the one-dimensional profile exponentially; see Del Pino-Kowalczyk-Pacard [21], Gui [39], and Kowalczyk-Liu-Pacard [48][49][50]. The existence of multiple-ended solutions and infinite-ended solutions to the Allen-Cahn equation in R 2 have been constructed in [2,22,51,52]. The structure and classification of four end solutions have been studied extensively in [21,40,[48][49][50].…”

Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on R^{N+1}

“…There is also extensive literature on solutions which are periodic in the first N variables and in the remaining variable they exhibit one or multiple transitions (homoclinic or heteroclinic behavior) between periodic solutions (see, for example, [31,39] and references therein). Solutions with symmetries instead of the periodicity in the first N variables have also been found and examined for elliptic equations and systems (see [3] and references therein).…”

The Heteroclinic Connection Problem for General Double-Well Potentials

“…In parallel, the interest in the heteroclinic connection problem stems also from the study of the vectorial Allen-Cahn equation that models multi-phase transitions (see [1], [3], [5], [6], [17], [22], and the references therein). Loosely speaking, the heteroclinic connections are expected to describe the way in which the solutions to the multi-dimensional parabolic system u t = ε 2 ∆u − ∇W (u), for small ε > 0, transition from one state to the other (see [21]).…”

“…If the global minima of W are non-degenerate, that is the Hessian ∂ 2 W (a ± ) is positive definite, the existence of a heteroclinic connection was proven in [49] by using techniques from Γ-convergence theory (an additional growth condition as |u| → ∞ was also assumed). Other variational proofs, which usually require some non-degeneracy of the global minima, can be found in [1], [2], [5], [22], [32] and [44]. In fact, as is pointed out, the proof of [2] carries over to the case where W vanishes to finite order at a ± .…”