Density estimates for degenerate double-well potentials

TL;DR

This paper establishes density estimates for phase coexistence models with degenerate double-well potentials, proving that near both phases the state parameters occupy significant space and that level sets converge uniformly to the interface.

Contribution

It introduces novel density estimates for degenerate potentials in phase coexistence models, applicable to Q-minima and general energy functionals without non-degeneracy assumptions.

Findings

Density estimates for degenerate double-well potentials.

Uniform convergence of level sets to the interface.

Applicability to quasilinear equations and Q-minima.

Abstract

We consider a general energy functional for phase coexistence models, which comprises the case of Banach norms in the gradient term plus a double-well potential. We establish density estimates for $Q$-minima. Namely, the state parameters close to both phases are proved to occupy a considerable portion of the ambient space. From this, we obtain the uniform convergence of the level sets to the limit interface in the sense of Hausdorff distance. The main novelty of these results lies in the fact that we do not assume the double-well potential to be non-degenerate in the vicinity of the minima. As far as we know, these types of density results for degenerate potentials are new even for minimizers and even in the case of semilinear equations, but our approach can comprise at the same time quasilinear equations, $Q$-minima and general energy functionals.