A few representation formulas for solutions of fractional Laplace equations - Djitté, Sidy Moctar (2025)
This paper is devoted to the Laplacian operator of fractional order s∈(0, 1) in several dimensions. We first establish a representation formula for the partial derivatives of the solutions of the homogeneous Dirichlet problem. Along the way, we obtain a Pohozaev-type identity for the fractional Green function and of the fractional Robin function. The latter extends to the fractional setting a formula obtained by Brezis and Peletier, see [11], in the classical case of the Laplacian. As an application we consider the particle system extending the classical point vortex system to the case of a fractional Laplacian. We observe that, for a single particle in a bounded domain, the properties of the fractional Robin function are crucial for the study of the steady states. We also extend the classical Hadamard variational formula to the fractional Green function as well as to the shape derivative of weak solution to the homogeneous Dirichlet problem. Finally we turn to the inhomogeneous Dirichlet problem and extend a formula by J.L. Lions, see [57], regarding the kernel of the reproducing kernel Hilbert space of harmonic functions to the case of s-harmonic functions. We observe that, despite the order of the operator is not 2, this formula looks like the Hadamard variational formula, answering in a negative way to an open question raised in [24
https://cris.fau.de/converis/publicweb/Publication/310944198
Keywords: Pohozaev identity; Hadamard formula; Green function; Robin function; Reproduction kernel Hilbert space; Point vortex system

Fractional Hardy–Rellich inequalities via integration by parts - De Nitti, Nicola; Djitté, Sidy Moctar (Nonlinear Analysis - Theory Methods & Applications, 2024)
We prove a fractional Hardy–Rellich inequality with an explicit constant in bounded domains of class C1,1. The strategy of the proof generalizes an approach pioneered by E. Mitidieri (Mat. Zametki, 2000) by relying on a Pohozaev-type identity.
CRIS-Team Scopus Importer:2024-03-15
Keywords: Fractional Lapalcian; Fractional Sobolev spaces; Hardy inequality; Pohozaev identity; Rellich inequality

Fractional Hardy-Rellich inequalities via integration by parts - Djitté, Sidy Moctar (2025)
We prove a fractional Hardy-Rellich inequality with an explicit constant in bounded domains of class C^1,1. The strategy of the proof generalizes an approach pioneered by E. Mitidieri (Mat. Zametki, 2000) by relying on a Pohozaev-type identity. <br
https://cris.fau.de/converis/publicweb/Publication/310944987
Keywords: Hardy inequality; Rellich inequality; Pohozaev identity; fractional Sobolev spaces; fractional Laplacian