Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

Abstract

We study the local Szegö-Weinberger profile in a geodesic ball Bg(y0, r0) centered at a point y0 in a Riemannian manifold (M, g). This profile is obtained by maximizing the first nontrivial Neumann eigenvalue µ2 of the Laplace-Beltrami Operator ∆g on M among subdomains of Bg(y0, r0) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of M at y0. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of ∆g, but additional difficulties arise due to the fact that µ2 is degenerate in the unit ball in R N and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.

Critical Domains for the First Nonzero Neumann Eigenvalue in Riemannian Manifolds

“…Next we restrict our attention to cylindrical manifolds of the type M := R k × N , where (N , g N ) is a closed connected manifold and the product metric g = g eucl ⊗ g N is considered on M . For the problem of maximizing µ 2 (Ω) among domains of fixed volume v, one may expect a different shape of maximizers depending on the size of v. If v > 0 is small, the results in [7] on the corresponding asymptotic profile expansion suggest that maximizing domains are perturbations of small geodesic ellipsoids in M , whereas for large v the domains…”