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 $\Omega\subset\mathbb{R}^N$ be a bounded $C^3$ domain. If $\Omega$ is a local minimizing domain for $\lambda_{s,p}(\Omega)=\inf{[u]_s^2:u\in\mathcal{H}*0^s(\Omega),\ |u|*{L^p(\Omega)}=1}$ under smooth volume-preserving deformations, then $\Omega$ is a ball. The proof first uses the fractional Hadamard formula to reduce shape minimality to the overdetermined boundary condition $u/\delta^s=C_0$. 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 $\rho=sp$, the boundary quotient is composed of finitely many constant-coefficient normal powers $\delta^{k\rho}$ and a $C^{1,\varepsilon}$ remainder; when $k\rho=1$, the unique resonant correction is $\delta\log\delta$. 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 $1<p<2$.