最佳分数阶Sobolev常数局部极小化区域的球刚性：次二次情形

中文摘要 AI 辅助

在《变分法与偏微分方程》60卷（2021年）第231篇论文的推论1.3之后，Djitte、Fall和Weth提出了一个问题：当1<p<2时，对于最佳分数阶Sobolev常数，满足体积约束的局部极小化区域是否仍必须是球。本文解决了该问题。设0<s<1，1<p<2，且Ω⊂R^N为有界C^3区域。若Ω在光滑保体积形变下是λ_{s,p}(Ω)=inf{[u]_s^2:u∈H_0^s(Ω), |u|_{L^p(Ω)}=1}的局部极小化区域，则Ω为球。证明首先利用分数阶Hadamard公式将形状极小性归结为超定边界条件u/δ^s=C_0。为克服因u^{p-1}在零点不满足Lipschitz条件而导致的移动平面法障碍，我们建立了一个加权奇异窄域极大值原理，其吸收因子恰为负集测度的sp/N次幂。角落处的困难通过有限边界展开解决：设ρ=sp，边界商由有限个常系数法向幂δ^{kρ}及C^{1,ε}余项组成；当kρ=1时，唯一的共振修正项为δ log δ。该展开使得正交角落两侧的所有低阶法向项相互抵消，从而得到移动平面角落引理所需要的一阶切向消失性。该方法不要求区域为凸，并覆盖了0<s<1和1<p<2的完整范围。

英文摘要

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 $1<p<2$.

发表机构

• Beijing Normal University（北京师范大学）

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