Density estimates for phase transitions with a trace

TL;DR

This paper introduces a modified Allen-Cahn model with a trace term, providing density estimates for interface level sets, and addresses degeneracy, singularity, and quasiminimal cases in a unified framework.

Contribution

It offers new density estimates for phase transition interfaces in a modified Allen-Cahn model, including degenerate and singular cases, unifying various scenarios.

Findings

Density estimates for level sets of interfaces

Unified treatment of degeneracy and singularity cases

Applicability to quasiminimal scenarios

Abstract

We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and singularity of the ellipticity of the model and the quasiminimal case.