On variational characterization of four-end solutions of the Allen–Cahn equation in the plane

Abstract

In this paper a novel variational method is developed to construct fourend solutions in R 2 for the Allen-Cahn equation. Four-end solutions have been constructed by Del Pino, Kolwazyck, Pacard and Wei when the angle θ of the ends is close to π/2 or 0 using the Lyapunov-Schmid reduction method, and later by Kolwazyck, Liu and Pacard using a continuation method for general θ ∈ (0, pi/2). By a special mountain pass argument in a restricted space, namely a set of monotone paths of monotone functions, a family of solutions in bounded domain is constructed with very good control of their nodal sets, which then leads to a four-end solution after sending the domains to R 2. In this way, not only a family of four end solutions is constructed for any angle θ ∈ (0π/2). The Morse index of such a four-end solution is also shown to be one.

Asymptotics for two-dimensional vectorial Allen-Cahn systems

“…Singularities x 0 which behave locally as ( 21)- (22) are termed of finite type. It turns out that singularities of finite type appear in the asymptotics of the vectorial Allen Cahn equation, even in the minimizing case, and are actually an intrinsic part in the problem.…”

“…Remark 2. Singularities of finite type have also be constructed as limits of scalar Allen-Cahn problem (see [17,22]). In these constructions, the number d of half-lines in (21) is even.…”

“…where the coordinates are chosen so that the tangent to S at x 0 has equation x 2 = 0, and where w ε stands for a solution to the one-dimension problem (23) (see Figure ??). Notice that the possibility of gluing of several such solutions is not excluded, but we will not discuss this here.…”

Two-end solutions to the Allen–Cahn equation in R3

“…The cross solution, constructed by Dang-Fife-Peletier ( [8]), represents a fourend solution with angle π 4 , while the almost parallel line solution, constructed by del Pino-Kowalczyk-Pacard-Wei ( [10]), represents a four-end solution with angle close to π 2 or 0. The existence of four-end solutions with any angle between 0 and π 2 was proved by two methods: the first through the moduli space theory by Kowalczyk-Liu-Pacard ( [19]), and the second approach by the mountain-pass variational method by us (Gui-Liu-Wei [15]). It was also shown that the fourend solutions have Morse index one ( [15]).…”

“…The existence of four-end solutions with any angle between 0 and π 2 was proved by two methods: the first through the moduli space theory by Kowalczyk-Liu-Pacard ( [19]), and the second approach by the mountain-pass variational method by us (Gui-Liu-Wei [15]). It was also shown that the fourend solutions have Morse index one ( [15]). On the other hand, monotone and symmetric properties of a general four end solution have been obtained in [14], where more general finite morse index solutions in R 2 have also been studied under an extra energy condition or a condition on the asymptotical structure of nodal curves.…”