On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

Vincenzo Ferone; Carlo Nitsch; Cristina Trombetti

doi:10.3934/cpaa.2015.14.63

Article Contents

2015, Volume 14, Issue 1: 63-82. Doi: 10.3934/cpaa.2015.14.63

This issue Previous Article On the validity of the Euler-Lagrange system Next Article Mean value properties of fractional second order operators

On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

1.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli
2.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli
3.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli

Received: January 31, 2014

Revised: January 31, 2014

Published: December 2014

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Abstract

For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are ``close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.

Keywords:

Sharp trace embeddings,
isoperimetric inequalities for eigenvalues,
weighted isoperimetric inequalities.

Mathematics Subject Classification: Primary: 46E35, 35P15; Secondary: 28A75.

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