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

Gloria Paoli, Gianpaolo Piscitelli, Leonardo Trani

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.

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

Abstract

We study, in dimension , the eigenvalue problem and the torsional rigidity for the -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.

Paper Structure

This paper contains 6 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Eigenvalue problems
  4. Torsional rigidity
  5. Quermassintegrals: definition and some properties
  6. Proof of the main result

Key Result

Proposition 2.2

Let $\beta\in\mathbb{R}\setminus \{0\}$. There exists a minimizer $u\in W^{1,p}(\Omega)$ of eigRN, which is a weak solution to case1.

Theorems & Definitions (18)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 8 more