Calculus of Variations and Geometric Measure Theory

D. Corona - S. Nardulli - R. Oliver-Bonafoux - G. Orlandi - P. Piccione

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

created by nardulli on 26 Feb 2024
modified by corona on 07 May 2025

[BibTeX]

Published Paper

Inserted: 26 feb 2024
Last Updated: 7 may 2025

Journal: Mathematische Annalen
Pages: 46
Year: 2025
Doi: https://doi.org/10.1007/s00208-025-03178-7

ArXiv: https://arxiv.org/pdf/2401.17847.pdf PDF
Links: Preprint

Abstract:

We present multiplicity results for mass constrained Allen-Cahn equations 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.