An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes

Francesco Della Pietra,Gianpaolo Piscitelli

doi:10.1007/s00032-020-00320-9

Abstract

In this paper we prove an optimal upper bound for the first eigenvalueof a Robin-Neumann boundary value problem for the p-Laplacian operator in domainswith convex holes. An analogous estimate is obtained for the correspondingtorsional rigidity problem.

Highlights

In this paper we prove an optimal bound for the first eigenvalue and the torsional rigidity of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes
We prove some properties of the first eigenvalue λ(β, Σ) and of the torsional rigidity T (β, Σ), as well as we recall some basic tool of convex analysis
We have proved that the first eigenvalue of the spherical shell is maximal among the domains with fixed volume and fixed (n − 1)-quermassintegral of the hole, while the torsional rigidity is minimal

Summary

Introduction

In this paper we prove an optimal bound for the first eigenvalue and the torsional rigidity of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. If the perimeter rather than the volume is fixed, the ball maximizes the first eigenvalue among all open, bounded, convex, smooth enough sets (see [2, 6]) In this framework, the case of Robin eigenvalue problems when β ≥ 0 is not a constant has been considered in [16], and, for example, in [4, 13, 11] where optimization with respect to β is considered. We conclude the paper with a section containing the possible future directions

Notation and preliminaries

Quermassintegrals and the Aleksandrov-Fenchel inequalities

Proof of main results

Conclusions