A few representation formulas for solutions of fractional Laplace equations

TL;DR

This paper develops new representation formulas and identities for solutions of fractional Laplace equations, extending classical results to the fractional setting and exploring applications in particle systems and shape optimization.

Contribution

It introduces new formulas for derivatives, Green functions, and shape derivatives for fractional Laplace equations, extending classical identities and addressing open questions.

Findings

Derived a representation formula for derivatives of solutions.

Extended Pohozaev-type identity to fractional Green functions.

Connected fractional Robin functions to steady states in particle systems.

Abstract

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ 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 Br\'ezis and Peletier, see \cite{Bresiz}, 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 in homogeneous Dirichlet problem and extend a formula by J.L. Lions, see \cite{Lions}, 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 \cite{ELPL}.