Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds

Abstract: The talk is concerned with geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains of a Riemannian manifold. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary value problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case where the first nonzero Neumann eigenvalue is not simple. In the second part of the talk, we focus on product manifolds with euclidean factors, and we classify the subdomains where the associated overdetermined boundary value problem has a solution. If time permits, I will also briefly discuss the first nontrivial Stekloff eigenvalue. This is joint work with Moustapha Fall (AIMS Senegal).

analysis of PDEs

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

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