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An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes

Francesco Della Pietra, Gianpaolo Piscitelli

Abstract

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes

Abstract

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

Paper Structure

This paper contains 7 sections, 7 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Eigenvalue problem
  4. The radial case
  5. The $p$-torsional rigidity
  6. Quermassintegrals and the Aleksandrov-Fenchel inequalities
  7. Proof of main results

Key Result

Theorem 1.1

Let $\beta>0$, and $\Omega$ and $D$ be two bounded open sets, with $D$ convex, $\Omega$ Lipschitz, connected, and $D\Subset \Omega$. Let be $\Sigma=\Omega\setminus \bar{D}$, $A=A_{R_1,R_2}=B_{R_{2}}\setminus \overline{B}_{R_{1}}$, where $B_{R_{i}}$ is a ball centered at the origin with radius $R_{i}

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof : Proof of Theorem \ref{['herschn']}
  • ...and 2 more