Pointwise Hadamard variational formula for the fractional Laplacian

We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-Δ)^s u = h$ in $Ω$ and $u=0$ in $Ω^c$, 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.

💡 Research Summary

The paper develops a pointwise Hadamard‑type variational formula for solutions of Dirichlet problems driven by the fractional Laplacian ((-\Delta)^s) with (0<s<1). The authors consider a bounded (C^{1,1}) domain (\Omega\subset\mathbb R^N) and a source term (h) that is either a Lipschitz function (defined on a larger set containing (\Omega)) or a Dirac delta distribution. For a smooth vector field (Y) they generate a family of diffeomorphisms (\Phi_t) with (\Phi_0=\mathrm{Id}) and (\partial_t\Phi_t|_{t=0}=Y); the perturbed domains are (\Omega_t=\Phi_t(\Omega)).

The main objects of study are:

1. The weak solution (u_t) of ((-\Delta)^s u_t = h) in (\Omega_t) with exterior zero condition.
2. The associated Green function (G_s^{\Omega_t}(x,y)) solving ((-\Delta)^s G = \delta_x) in (\Omega_t) and vanishing outside.
3. The Robin function (R_s^{\Omega}=H_s^{\Omega}(x,x)) where (G_s^{\Omega}=F_s-H_s^{\Omega}) and (F_s) is the fundamental solution.

The authors distinguish between the Lagrangian shape derivative (v’=\partial_t(u_t\circ\Phi_t)|{t=0}) and the Eulerian shape derivative (u’ = v’ - \nabla u\cdot Y). The Eulerian derivative is the physically relevant quantity because it coincides with the pointwise limit (\lim{t\to0}(u_t(x)-u(x))/t).

A crucial technical tool is the definition of a fractional Neumann trace on the boundary: \