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.

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

References in corpus (5)

• On the Multiplicity One Conjecture in Min-max theory
• Multiplicity one and strictly stable Allen-Cahn minimal hypersurfaces
• The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces
• Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry
• Lusternik-Schnirelman and Morse theory for the Van der Waals-Cahn-Hilliard equation with volume constraint

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