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Symmetry of fractional Neumann eigenfunctions in the ball

Vladimir Bobkov, Enea Parini

Abstract

We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian $(-Δ)^s$, where $s \in (0,1)$, in an $N$-dimensional ball with nonlocal Neumann boundary conditions. By means of a spectral stability result, we prove that, when $s$ is sufficiently close to $1$, the eigenspace associated to the first nontrivial eigenvalue is generated by $N$ antisymmetric eigenfunctions with exactly two nodal domains in the ball.

Symmetry of fractional Neumann eigenfunctions in the ball

Abstract

We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian , where , in an -dimensional ball with nonlocal Neumann boundary conditions. By means of a spectral stability result, we prove that, when is sufficiently close to , the eigenspace associated to the first nontrivial eigenvalue is generated by antisymmetric eigenfunctions with exactly two nodal domains in the ball.
Paper Structure (6 sections, 15 theorems, 100 equations)

This paper contains 6 sections, 15 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Functional setting
  4. Neumann eigenvalue problem
  5. Spectral stability for $s \to 1^-$
  6. Symmetry results

Key Result

Theorem 1.1

Let $B \subset {\mathbb R}^N$ be a ball centered at the origin. Let $s \in (0,1)$. Then the following dichotomy occurs:

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 24 more