<p>This paper is devoted to the Laplacian operator of fractional order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$s\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in several dimensions. We consider the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(-\Delta )^su=f(x,u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega ^c$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Ω</mi> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation> and establish a representation formula for partial derivatives of solutions in terms of the normal derivative <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u/\delta ^s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">/</mo> <msup> <mi>δ</mi> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. As a consequence, we prove that solutions to the overdetermined problem <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(-\Delta )^su=f(x,u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$u=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Omega ^c$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Ω</mi> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$u/\delta ^s=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">/</mo> <msup> <mi>δ</mi> <mi>s</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> are globally Lipschitz continuous provided that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$2s>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 Brezis and Peletier (<CitationRef CitationID="CR12">1989</CitationRef>) 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.</p>

A Brezis and Peletier Type Result for the Fractional Robin Function

期刊论文
发表日期：

2025年12月11日

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Sidy M. Djitte,
Franck Sueur

摘要

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ $s \in (0, 1)$ in several dimensions. We consider the equation $(-\Delta )^su=f(x,u)$ ${(- Δ)}^{s} u = f (x, u)$ in $\Omega $ $Ω$ , $u=0$ $u = 0$ in $\Omega ^c$ $Ω^{c}$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/\delta ^s$ $u / δ^{s}$ . As a consequence, we prove that solutions to the overdetermined problem $(-\Delta )^su=f(x,u)$ ${(- Δ)}^{s} u = f (x, u)$ in $\Omega $ $Ω$ , $u=0$ $u = 0$ in $\Omega ^c$ $Ω^{c}$ , and $u/\delta ^s=0$ $u / δ^{s} = 0$ on $\partial \Omega $ $\partial Ω$ are globally Lipschitz continuous provided that $2s>1$ $2 s > 1$ . 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 Brezis and Peletier (1989) 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.