Faber-Krahn inequality for Robin problem involving p-Laplacian

TL;DR

This paper proves a Faber-Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions, showing that the ball minimizes the eigenvalue among all domains of fixed volume.

Contribution

It establishes a Faber-Krahn inequality for the Robin p-Laplace eigenvalue problem, extending classical results to this nonlinear and boundary condition setting.

Findings

The ball minimizes the first eigenvalue for fixed volume domains.

The inequality holds for the p-Laplace operator with Robin boundary conditions.

The result generalizes known inequalities for Laplacian operators.

Abstract

The eigenvalue problem for the p-Laplace operator with Robin boundary condition is considered in this paper. A Faber-Krahn type inequality is proved. More precisely, it is shown that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue.