Pointwise Hadamard variational formula for the fractional Laplacian

TL;DR

This paper derives pointwise shape derivative formulas for solutions to fractional Laplacian problems, extending classical Hadamard formulas to nonlocal operators using PDE techniques.

Contribution

It provides the first pointwise shape derivative formulas for fractional Laplacian solutions, involving boundary integrals and fractional Neumann traces.

Findings

Solutions are shape differentiable in all directions.

Derived explicit formulas involving boundary integrals.

Extended classical Hadamard formulas to fractional Laplacian.

Abstract

We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-\Delta)^s u = h$ in $\Omega$ and $u=0$ in $\Omega^c$, where the right-hand side $h$ is either a Dirac delta distribution or a Lipschitz function. In both cases, we prove that the corresponding solution is shape differentiable in every direction and we derive a formula for the pointwise value of its shape derivative. These formulas involve integral on the domain's boundary and fractional Neumann's traces. This extends to the case of the fractional Laplacian the well-known Hadamard variational formula for the standard Laplacian. Our argument is in the spirit of \cite{Ushikoshi, Kozono-Ushikoshi} and is based on PDEs techniques.