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A nonlocal anisotropic eigenvalue problem

Gianpaolo Piscitelli

Abstract

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq twisted\rq\rq problem, we show that, this problem displays a \emph{saturation} phenomenon: the first eigenvalue increases with the weight up to a critical value and then remains constant.

A nonlocal anisotropic eigenvalue problem

Abstract

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq twisted\rq\rq problem, we show that, this problem displays a \emph{saturation} phenomenon: the first eigenvalue increases with the weight up to a critical value and then remains constant.

Paper Structure

This paper contains 8 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gauge functions
  4. Rearrangements
  5. Properties of local problem
  6. The first eigenvalue of the nonlocal problem
  7. On the First Twisted Dirichlet Eigenvalue
  8. The Nonlocal Problem

Key Result

Theorem 1.1

For every $n\geq 2$, there exists a positive value such that, for every bounded, open set $\Omega$ in $\mathbb{R}^n$ and for every real number $\alpha$, it holds If equality sign holds when $\alpha |\Omega|^{1+2/n}<\alpha_c$ then $\Omega$ is a Wulff set, while if inequality sign holds when $\alpha|\Omega|^{1+2/n}>\alpha_c$ then $\Omega$ is the union of two disjoint Wulff sets of equal measure.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 16 more