{"title":"On the Triple Junction Problem without Symmetry Hypotheses","authors":"Nicholas D. Alikakos, Zhiyuan Geng","doi":"10.1007/s00205-024-01966-0","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate the Allen–Cahn system <span>\\(\\Delta u-W_u(u)=0\\)</span>, <span>\\(u:\\mathbb {R}^2\\rightarrow \\mathbb {R}^2\\)</span>, where <span>\\(W\\in C^2(\\mathbb {R}^2,[0,+\\infty ))\\)</span> is a potential with three global minima. We establish the existence of an entire solution <i>u</i> which possesses a triple junction structure. The main strategy is to study the global minimizer <span>\\(u_\\varepsilon \\)</span> of the variational problem <span>\\(\\min \\int _{B_1} \\left( \\frac{\\varepsilon }{2}\\vert \\nabla u\\vert ^2+\\frac{1}{\\varepsilon }W(u) \\right) \\,\\textrm{d}z\\)</span>, <span>\\(u=g_\\varepsilon \\)</span> on <span>\\(\\partial B_1\\)</span> for some suitable boundary data <span>\\(g_\\varepsilon \\)</span>. The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution or on the potential.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-01966-0","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the Allen–Cahn system \(\Delta u-W_u(u)=0\), \(u:\mathbb {R}^2\rightarrow \mathbb {R}^2\), where \(W\in C^2(\mathbb {R}^2,[0,+\infty ))\) is a potential with three global minima. We establish the existence of an entire solution u which possesses a triple junction structure. The main strategy is to study the global minimizer \(u_\varepsilon \) of the variational problem \(\min \int _{B_1} \left( \frac{\varepsilon }{2}\vert \nabla u\vert ^2+\frac{1}{\varepsilon }W(u) \right) \,\textrm{d}z\), \(u=g_\varepsilon \) on \(\partial B_1\) for some suitable boundary data \(g_\varepsilon \). The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution or on the potential.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.