A Brezis and Peletier Type Result for the Fractional Robin Function

[en] This paper is devoted to the Laplacian operator of fractional order s∈(0,1) in several dimensions. We consider the equation (-Δ)su=f(x,u) in Ω, u=0 in Ωc and establish a representation formula for partial derivatives of solutions in terms of the normal derivative u/δs. As a consequence, we prove that solutions to the overdetermined problem (-Δ)su=f(x,u) in Ω, u=0 in Ωc, and u/δs=0 on ∂Ω 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 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.

Precision for document type :

Review article

Disciplines :

Mathematics

Author, co-author :

DJITTE, Sidy Moctar ^✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

SUEUR, Franck ^✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

^✱ These authors have contributed equally to this work.

External co-authors :

Language :

English

Title :

A Brezis and Peletier Type Result for the Fractional Robin Function

Publication date :

2026

Journal title :

Potential Analysis

ISSN :

0926-2601

eISSN :

1572-929X

Publisher :

Springer

Volume :

Issue :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.springer.com/content/pdf/10.1007/s11118-025-10240-1.pdf

Funding text :

The first author was partially supported by the Alexander von Humboldt-Professorship program and by the Transregio 154 Project \u201CMathematical Modelling, Simulation and Optimization Using the Example of Gas Networks\u201D of the Deutsche Forschungsgemeinschaft.

Commentary :

21 pages

Available on ORBilu :

since 10 February 2026

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