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
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 LusternikāSchnirelmann 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.