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A Brezis and Peletier type result for the fractional Robin function

Sidy M. Djitte, Franck Sueur

TL;DR

This work extends classical Brézis–Peletier-type results to the fractional Laplacian. It develops a representation formula for the spatial derivatives of solutions to $(-\Delta)^s u=f(x,u)$ in a bounded domain in terms of a boundary normal trace $u/\delta^s$ and a volume term, valid with a threshold at $2s>1$ that affects regularity. It then derives a Pohozaev-type identity for the fractional Green function and a gradient representation for the fractional Robin function, including a boundary-integral formula and symmetry-driven consequences; these yield nondegeneracy and critical-point results for symmetric domains. The methods hinge on smooth approximations of the Green function, a fractional product rule, and careful limiting arguments to pass from regularized identities to exact derivative formulas. The results deepen fractional potential theory, providing tools for boundary behavior, domain geometry, and concentration phenomena in nonlocal elliptic problems.

Abstract

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/δ^s$. As a consequence, we prove that solutions to the overdetermined problem $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$, and $u/δ^s=0$ on $\partialΩ$ are globally Lipschitz continuous provided that $2s>1$. We also prove a Pohozaev-type identity for the Green function and, in particular, obtain a formula for the gradient of the Robin function, which extends to the fractional setting some results obtained by Brézis and Peletier in \cite{Bresiz} in the classical case of the Laplacian. Finally, an application to the nondegeneracy of critical points of the fractional Robin function in symmetric domains is discussed.

A Brezis and Peletier type result for the fractional Robin function

TL;DR

This work extends classical Brézis–Peletier-type results to the fractional Laplacian. It develops a representation formula for the spatial derivatives of solutions to in a bounded domain in terms of a boundary normal trace and a volume term, valid with a threshold at that affects regularity. It then derives a Pohozaev-type identity for the fractional Green function and a gradient representation for the fractional Robin function, including a boundary-integral formula and symmetry-driven consequences; these yield nondegeneracy and critical-point results for symmetric domains. The methods hinge on smooth approximations of the Green function, a fractional product rule, and careful limiting arguments to pass from regularized identities to exact derivative formulas. The results deepen fractional potential theory, providing tools for boundary behavior, domain geometry, and concentration phenomena in nonlocal elliptic problems.

Abstract

This paper is devoted to the Laplacian operator of fractional order in several dimensions. We consider the equation in , in and establish a representation formula for partial derivatives of solutions in terms of the normal derivative . As a consequence, we prove that solutions to the overdetermined problem in , in , and on are globally Lipschitz continuous provided that . We also prove a Pohozaev-type identity for the Green function and, in particular, obtain a formula for the gradient of the Robin function, which extends to the fractional setting some results obtained by Brézis and Peletier in \cite{Bresiz} in the classical case of the Laplacian. Finally, an application to the nondegeneracy of critical points of the fractional Robin function in symmetric domains is discussed.
Paper Structure (16 sections, 14 theorems, 131 equations)

This paper contains 16 sections, 14 theorems, 131 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Representation formula for partial derivatives
  4. Representation formulas for gradient of the fractional Robin function
  5. Proof of Theorem \ref{['main-result']}
  6. Scheme of the proof
  7. Proof of Lemma \ref{['Lem6.1']}
  8. Proof of Lemma \ref{['Lem4.2']}
  9. Proof of Lemma \ref{['ess-convergence-result']}
  10. Proof of Lemma \ref{['Lem4.3']}
  11. Proof of Theorem \ref{['Thm.2']}
  12. Proof of \ref{['partial-Green']}
  13. Proof of Lemma \ref{['lem-lim-G-x-y']}
  14. Proof of Corollary \ref{['nodeg']}
  15. Proof of Corollary \ref{['non-deg-frac-Robin']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $N> 2s$ and let $\Omega$ be a bounded open set of $\mathbb{R}^N$ of class $C^{1,1}$ and $f:\mathbb{R}^N\times\mathbb{R}\to \mathbb{R}$ satisfies H. Let $u$ be a weak solution of sect3-H. Then, for all $x\in \Omega$, we have In the case $2s\leqslant 1$, we have In particular, in the case $(-\Delta)^su=\text{const}$ in $\Omega$ & $u=0$ in $\mathbb{R}^N\setminus\Omega$, we have

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 12 more