Lusternik–Schnirelman and Morse Theory for the Van der Waals–Cahn–Hilliard equation with volume constraint
Vieri Benci, Stefano Nardulli, Luis Eduardo Osorio Acevedo, Paolo Piccione · Nonlinear Analysis · 2022
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 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.