Gradient flow of phase transitions with fixed contact angle

TL;DR

This paper analyzes the gradient flow of the Allen-Cahn equation with fixed contact angles, establishing convergence properties and boundary conditions, and deriving a monotonicity formula for energy measures near boundaries.

Contribution

It provides new convergence results and boundary condition characterizations for Allen-Cahn gradient flows with fixed contact angles, including an Ilmanen type monotonicity formula.

Findings

Solutions converge both inside the domain and at the boundary.

Limiting varifolds satisfy generalized contact angle conditions.

An Ilmanen type monotonicity formula is established for energy measures.

Abstract

We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary.