From Neumann to Steklov and beyond, via Robin: the Weinberger way

TL;DR

This paper investigates the maximization of the second Robin Laplacian eigenvalue for domains of fixed volume, showing the ball's optimality in certain parameter regimes and exploring the transition between Neumann and Steklov eigenvalues.

Contribution

It establishes the ball as the maximizer of the second Robin Laplacian eigenvalue for negative parameters within a specific regime, extending understanding of eigenvalue optimization.

Findings

The ball maximizes the second Robin eigenvalue for negative parameters in a certain regime.

Maximality of the ball fails for sufficiently large negative parameters.

No maximizer exists when the Robin parameter is positive.

Abstract

The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative, and the problem admits no maximiser when $\alpha$ is positive.