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A monotonicity result for the first Steklov-Dirichlet Laplacian eigenvalue

Nunzia Gavitone, Gianpaolo Piscitelli

Abstract

In this paper, we consider the first Steklov-Dirichlet eigenvalue of the Laplace operator in annular domain with a spherical hole. We prove a monotonicity result with respect the hole, when the outer region is centrally symmetrc.

A monotonicity result for the first Steklov-Dirichlet Laplacian eigenvalue

Abstract

In this paper, we consider the first Steklov-Dirichlet eigenvalue of the Laplace operator in annular domain with a spherical hole. We prove a monotonicity result with respect the hole, when the outer region is centrally symmetrc.

Paper Structure

This paper contains 6 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction and main result
  2. The Steklov-Dirichlet Laplacian eigenvalue problem
  3. Foundation of the problem
  4. Differentiability of $\sigma(t)$
  5. The first order Shape Derivative of $\sigma(t)$
  6. The second order Shape Derivative of $\sigma(t)$

Key Result

Theorem 1.2

Let $\Omega_0$ be as in Definition set, $w \in \mathbb{R}^n$ be a unit vector, $B_r(t)$, $\rho_w$, $\Omega(t)$ and $\sigma(t)$ be defined as in ph, row, sett and eigt, respectively. Then, $\sigma( t)$ is strictly monotone decreasing with respect to $t\in [0,\rho_w-r)$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 5 more