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Kohler-Jobin inequality for $p$-Laplace operator in the Gauss space

Francesco Chiacchio, Vincenzo Ferone, Anna Mercaldo, Jing Wang

Abstract

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized torsional rigidity and on a sharp mass comparison result. Furthermore, a Payne-Rayner type inequality is established.

Kohler-Jobin inequality for $p$-Laplace operator in the Gauss space

Abstract

A sharp lower bound for the first Dirichlet eigenvalue of the -laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized torsional rigidity and on a sharp mass comparison result. Furthermore, a Payne-Rayner type inequality is established.

Paper Structure

This paper contains 9 sections, 16 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Symmetrization with respect to Gauss measure
  4. First eigenvalue and torsional rigidity
  5. A generalized torsional rigidity
  6. Comparison results
  7. Main result
  8. An explicit solution when p=2
  9. Appendix

Key Result

Theorem 2.1

Let $p>1$ and $u\in W^{1,p}_0(\Omega,\phi_N )$. Then $u^\sharp \in W^{1,p}_0(\Omega^\sharp,\phi_N)$ and

Theorems & Definitions (30)

  • Theorem 2.1
  • Proposition 2.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 20 more