Sharp estimates for the first $p$-Laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes

TL;DR

This paper establishes sharp bounds for the first p-Laplacian eigenvalue and p-torsional rigidity on convex sets with holes, showing extremal properties of annuli under fixed measure and perimeter.

Contribution

It provides new extremal inequalities for eigenvalues and torsional rigidity of p-Laplacian on convex sets with holes, with precise geometric conditions.

Findings

Annuli maximize the first eigenvalue under fixed measure and perimeter.

Annuli minimize the p-torsional rigidity under the same conditions.

Results hold for convex sets with holes in dimensions n ≥ 2.

Abstract

We study, in dimension $n\geq2$, the eigenvalue problem and the torsional rigidity for the $p$-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus maximizes the first eigenvalue and minimizes the torsional rigidity when the measure and the external perimeter are fixed.