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<p>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 category 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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月7日

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

摘要

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 category 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.