Variational aspects of phase transitions with prescribed mean curvature

TL;DR

This paper investigates the spectral properties of phase transitions with prescribed mean curvature in Riemannian manifolds, providing bounds and asymptotics for eigenvalues of related elliptic PDEs and analyzing the geometric limits of diffuse interfaces.

Contribution

It introduces new bounds and asymptotic estimates for eigenvalues of phase transition problems with prescribed mean curvature, advancing understanding of their geometric and spectral behavior.

Findings

Established upper bounds for eigenvalues of the diffuse problem.

Derived sharp asymptotics for eigenvalues with multiplicity one.

Provided $C^{2,eta}$ estimates on phase transition layers.

Abstract

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asymptotics that are sharp to order $o(\varepsilon^2)$ and $C^{2,\alpha}$ estimates on multiplicity-one phase transition layers.