Sharp lower bounds for vector Allen-Cahn energy and qualitative properties of minimizes under no symmetry hypotheses

2021/10/01 by Nicholas D. Alikakos, Alikakos, Nicholas D., Giorgio Fusco +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2110.00388

openalex publication_date 2021/10/01 · openalex created_date 2021/10/11 · openalex updated_date 2026/07/28

Abstract

We study vector minimizers u of the Allen-Cahn functional with potentials possessing N global minima defined on bounded domains, with certain geometrical features and Dirichlet conditions on the boundary. We derive a sharp lower bound for the energy (as ε→ 0) with the additional feature that it involves half of the gradient and part of the domain. Based on this we derive very precise (in ε) pointwise estimates up to the boundary for uε. Depending on the geometry of the domain uε exhibits either boundary layers or internal layers. We do not impose symmetry hypotheses.

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