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⊂ℝN be a bounded C3 domain. If Ω is a local minimizing domain for λs,p(Ω)=inf[u]s2:u∈ℋ*0s(Ω), |u|*Lp(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/δs=C0. To overcome the moving-plane obstruction caused by the failure of up−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 δkρ and a C1,ε remainder; when kρ=1, the unique resonant correction is δlogδ. 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.
Related
- Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
- Comparison principles and symmetry for subquadratic fractional p-Laplacian equations
- Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
- Outward minimizing p-capacity, horizons, and Schwarzschild rigidity
- Rigidity and stability for the Bogovskii constant
BibTeX
@misc{deng2026ball,
title = {{Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case}},
author = {Zikang Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13215},
eprint = {2609.13215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13215},
}
RIS
TY - UNPB
TI - Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case
AU - Zikang Deng
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.13215
UR - https://arxiv.org/abs/2609.13215
AB - 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⊂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∈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,ε remainder; when kρ=1, the unique resonant correction is δlogδ. 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.
ER -