Variational aspects of phase transitions with prescribed mean curvature

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 $o({\epsilon }^2)$ and $C^{2,\alpha }$ estimates on multiplicity-one phase transition layers.

arXiv commentTo appear in CVPDE. Changes from previous version: some a priori background smoothness assumptions were increased to allow for a simplified regularity theory presentation

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Cited by in corpus (2)

• A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory
• Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds