Linear stability implies nonlinear stability for Faber–Krahn-type inequalities
Mark Allen
Brigham Young University, Provo, USADennis Kriventsov
Rutgers University, Piscataway, USARobin Neumayer
Carnegie Mellon University, Pittsburgh, USA
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Abstract
For a domain and a small number , let
be a modification of the first Dirichlet eigenvalue of . It is well known that over all with a given volume, the only sets attaining the infimum of are balls ; this is the Faber–Krahn inequality. The main result of this paper is that, if for all with the same volume and barycenter as whose boundaries are parametrized as small normal graphs over with bounded norm
(i.e., the Faber–Krahn inequality is linearly stable), then the same is true for any with the same volume and barycenter as without any smoothness assumptions (i.e., it is nonlinearly stable). Here stands for an normalized first Dirichlet eigenfunction of . Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.
Cite this article
Mark Allen, Dennis Kriventsov, Robin Neumayer, Linear stability implies nonlinear stability for Faber–Krahn-type inequalities. Interfaces Free Bound. 25 (2023), no. 2, pp. 217–324
DOI 10.4171/IFB/487