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A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle

Gloria Paoli, Gianpaolo Piscitelli, Rossano Sannipoli

Abstract

In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherichal obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle

Abstract

In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherichal obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

Paper Structure

This paper contains 7 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The Eigenvalue Problem
  3. Eigenvalues and Eigenfunctions
  4. A first upper bound
  5. Volume constraint on the spherical shells
  6. Spherical shell with fixed difference between radii.
  7. Main result

Key Result

Proposition 2.2

Let $r>0$ and $\Omega\in\mathcal{A}_r$, then there exists a function $u\in H^1_{\partial B_{r}}(\Omega)$ achieving the minimum in minSD and satisfying problem eigSD. Moreover, $u$ is positive (or negative) in $\Omega$.

Theorems & Definitions (19)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 9 more