Calculus of Variations and Geometric Measure Theory

V. Benci - D. Corona - S. Nardulli - L. E. Osorio Acevedo - P. Piccione

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

created by nardulli on 14 Jul 2020
modified on 26 Feb 2024


Published Paper

Inserted: 14 jul 2020
Last Updated: 26 feb 2024

Journal: Nonlinear Analysis
Pages: 30
Year: 2022
Doi: 10.1016/

ArXiv: 2007.07024 PDF
Links: Published version


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.