Sharp spectral bounds on starlike domains

TL;DR

This paper establishes sharp spectral bounds for starlike domains, showing the ball optimizes certain eigenvalues and spectral functions, with results applicable to various boundary conditions and domain shapes.

Contribution

It introduces new sharp bounds on Laplacian eigenvalues incorporating geometric deviation measures, extending classical inequalities to broader domain classes.

Findings

The ball maximizes the first eigenvalue.

The ball minimizes the spectral zeta function and heat trace.

Results apply to convex and starlike domains with various boundary conditions.

Abstract

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber--Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by P\'{o}lya and Szeg\H{o}. Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.