References & Citations
Mathematics > Analysis of PDEs
Title: Allen-Cahn Solutions with Triple Junction Structure at Infinity
(Submitted on 22 May 2023 (v1), last revised 25 Apr 2024 (this version, v2))
Abstract: We construct an entire solution $U:\mathbb{R}^2\to\mathbb{R}^2$ to the elliptic system \[ \Delta U=\nabla_uW(U), \] where $W:\mathbb{R}^2\to [0,\infty)$ is a `triple-well' potential. This solution is a local minimizer of the associated energy \[ \int \frac{1}{2}|\nabla U|^2+W(U)\,dx \] in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set. Furthermore, we show that along subsequences, the `blowdowns' of $U$ given by $U_R(x):=U(Rx)$ approach a minimal triple junction as $R\to\infty$. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions.
Submission history
From: Peter Sternberg [view email][v1] Mon, 22 May 2023 20:39:31 GMT (101kb,D)
[v2] Thu, 25 Apr 2024 20:41:08 GMT (103kb,D)
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