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Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

Michele Gatti, Julian Scheuer, Tobias Weth

Abstract

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $Ω\subseteq \mathbb{R}^n$. Specifically, we show that if $\overlineΩ$ has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to $\partial Ω$, then $Ω$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $Ω$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $Ω$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.

Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

Abstract

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set . Specifically, we show that if has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to , then must be a ball. Remarkably, this conclusion remains valid under the sole assumption that is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on . Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.

Paper Structure

This paper contains 12 sections, 14 theorems, 185 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Symmetry results.
  3. Stability results.
  4. Structure of the paper.
  5. Preliminaries
  6. Fractional setting.
  7. Remarks on solutions to the overdetermined problems.
  8. The tangent and normal cones.
  9. Proof of the symmetry result for the torsion problem
  10. Proof of the stability result for the torsion problem
  11. Generalization of the symmetry and stability results
  12. Further stability results

Key Result

Theorem 1.1

Let $\Omega \subseteq \mathbb{R}^n$ be a regular open, bounded set whose closure $\overline{\Omega}$ has positive reach $r_{\Omega} > 0$, and let $t \in (0,r_\Omega)$. If the unique solution $u\in H^{s}(\mathbb{R}^{n})$ of the fractional torsion problem eq:mainprob-frac satisfies the overdetermined with some constant $c_\flat \in \mathbb{R}$, then $\Omega$ is a ball.

Figures (2)

  • Figure 1: The plane stops upon touching $\Omega$, so $\Omega_\star = \varnothing$. At the critical position, both \ref{['it:int-touch']} and \ref{['it:non-transv-inter']} occur simultaneously.
  • Figure 2: An example in which our assumptions on $\Omega$ and $G$ are violated.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Remark 3.1
  • ...and 18 more