An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

TL;DR

This paper proves that among symmetric bounded domains with a fixed weighted measure, the ball maximizes the second Robin eigenvalue of the weighted Laplacian for certain negative Robin parameters, advancing shape optimization understanding.

Contribution

It establishes a new isoperimetric inequality for the second Robin eigenvalue of the weighted Laplacian on symmetric domains, identifying the ball as the optimizer.

Findings

The ball maximizes the second Robin eigenvalue among symmetric domains.

The result applies for a range of negative Robin parameters.

The theorem extends shape optimization results to weighted Laplacians with Robin boundary conditions.

Abstract

In this paper, we investigate a shape optimization problem for the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains symmetric about the origin. Our main theorem states that the ball centered at the origin maximizes the second Robin eigenvalue among all Lipschitz bounded domains of prescribed weighted measure and symmetric about the origin for a range of negative Robin parameters.