A Brezis and Peletier type result for the fractional Robin function

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/δ^s$. As a consequence, we prove that solutions to the overdetermined problem $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$, and $u/δ^s=0$ on $\partialΩ$ 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 Brézis and Peletier in \cite{Bresiz} 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.

💡 Research Summary

The paper investigates the fractional Laplacian (−Δ)^s with order s∈(0,1) in a bounded C^{1,1} domain Ω⊂ℝ^N (N≥1). The authors consider the Dirichlet problem

 (−Δ)^s u = f(x,u) in Ω, u = 0 in ℝ^N\Ω,

where the non‑linearity f satisfies a mild regularity hypothesis (C^{β}{loc} if 2s>1, C^{1,β}{loc} otherwise). They first recall the classical representation of the solution via the Green function G_s(x,z):

 u(x)=∫_Ω G_s(x,z) f(z,u(z)) dz,

and note that G_s splits into a singular fundamental solution F_s and a regular part H_Ω solving a homogeneous exterior problem.

The main technical achievement is a representation formula for the spatial derivatives of u in terms of boundary data involving the fractional normal derivative γ_s^0(w)=w/δ^s|_{∂Ω}, where δ denotes the distance to the boundary. In the integer‑order case one can differentiate under the integral sign and obtain a simple formula using Green’s third identity. For the fractional case the singularity of G_s prevents a direct differentiation. The authors overcome this by introducing two families of smooth cut‑off functions:

1. ξ_k(y)=1−ρ(k δ(y)), which vanishes near the boundary, and
2. ϕ_{μ,x}(y)=1−ρ(8|x−y|^2/δ(x)^2 μ^2), which removes a neighbourhood of the pole x.

Here ρ∈C_c^∞(−2,2) equals 1 on (−1,1). The product ξ_k ϕ_{μ,x} G_s(x,·) is smooth and compactly supported, allowing it to be used as a test function in the fractional integration‑by‑parts identity

 ∫_Ω ∂_i v (−Δ)^s w dz = −∫_Ω ∂i w (−Δ)^s v dz − Γ(1+s)^2 ∫{∂Ω} γ_s^0(v) γ_s^0(w) ν_i dσ.

Applying this identity with v=ξ_k ϕ_{μ,x} G_s(x,·) and w=ξ_k u, expanding the fractional Laplacian of a product, and passing to the limits k→∞, μ→0, the authors obtain the following formulas.

If 2s>1 (so that the Green function gradient is integrable),

 ∂{x_i}u(x)=−Γ(1+s)^2 ∫{∂Ω} γ_s^0(u) γ_s^0(G_s(x,·)) ν_i dσ − ∫Ω ∂{z_i}G_s(x,z) f(z,u(z)) dz. (7)

If 2s≤1, an additional term involving the derivatives of f appears:

 ∂{x_i}u(x)=−Γ(1+s)^2 ∫{∂Ω} γ_s^0(u) γ_s^0(G_s(x,·)) ν_i dσ
  + ∫_Ω G_s(x,z)