Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case

Abstract

After Corollary 1.3 in Calculus of Variations and Partial Differential Equations 60 (2021), Paper 231, Djitte, Fall, and Weth asked whether, when $1<p<2$, a volume-constrained local minimizing domain for the best fractional Sobolev constant must still be a ball. This paper solves that problem. Let $0<s<1$, $1<p<2$, and let $Ω\subset\mathbb{R}^N$ be a bounded $C^3$ domain. If $Ω$ is a local minimizing domain for $λ_{s,p}(Ω)=\inf{[u]_s^2:u\in\mathcal{H}*0^s(Ω),\ |u|*{L^p(Ω)}=1}$ under smooth volume-preserving deformations, then $Ω$ is a ball. The proof first uses the fractional Hadamard formula to reduce shape minimality to the overdetermined boundary condition $u/δ^s=C_0$. To overcome the moving-plane obstruction caused by the failure of $u^{p-1}$ to be Lipschitz at zero, we establish a weighted singular narrow-domain maximum principle whose absorption factor is exactly the $sp/N$ power of the measure of the negative set. The difficulty at a corner is resolved by a finite boundary expansion: setting $ρ=sp$, the boundary quotient is composed of finitely many constant-coefficient normal powers $δ^{kρ}$ and a $C^{1,\varepsilon}$ remainder; when $kρ=1$, the unique resonant correction is $δ\logδ$. This expansion makes all lower-order normal terms on the two sides of an orthogonal corner cancel, thereby yielding the first-order tangential vanishing required by the moving-plane corner lemma. The method does not require the domain to be convex and covers the full range $0<s<1$ and $1<p<2$.