Submitted Paper

Inserted: 26 feb 2024
Last Updated: 16 mar 2024

Journal: ArXiv
Pages: 48
Year: 2024

Abstract:

We present multiplicity results for mass constrained Allen-Cahn equa- tions on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik–Schnirelmann cat- egory of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non- degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.