On the $ \Gamma $-convergence of the Allen-Cahn functional with boundary conditions

TL;DR

This paper analyzes the asymptotic behavior of minimizers of the Allen-Cahn functional with boundary conditions, establishing the $ ext{Gamma}$-convergence to a limit related to minimal partitions and geometric structures.

Contribution

It provides a rigorous $ ext{Gamma}$-convergence result for the Allen-Cahn functional with boundary conditions and characterizes the structure of minimizers in the limit.

Findings

$ ext{Gamma}$-limit of the Allen-Cahn functional with boundary conditions is established.

Minimizers of the limit functional correspond to minimal partitions of the domain.

The structure of minimizers in the limit is characterized using minimal cones and regularity of minimal curves.

Abstract

We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $ \varepsilon $-energy functional as $ \varepsilon \rightarrow 0 $.