Lusternik-Schnirelman and Morse theory for the Van der Waals-Cahn-Hilliard equation with volume constraint

Abstract

We give a multiplicity result for solutions of the Van der Waals-Cahn-Hilliard two-phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results from the classical Lusternik--Schnirelman and Morse theory, together with a technique, the so-called \emph{photography method}, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.

The second author appears only in the Corrigendum. 29 pages, 5 figures. Accept in Nonlinear Analysis, 2022

Cited by in corpus (2)

• Variational aspects of phase transitions with prescribed mean curvature
• Generic nondegeneracy for solutions of the Allen-Cahn equation under a volume constraint in closed manifolds