发表机构
Politecnico di Bari; EPFL, Institut de Mathématiques; Universität Bielefeld（巴里理工大学; 洛桑联邦理工学院数学研究所; 比勒费尔德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该数学研究针对分数拉普拉斯算子的正基态，证明其满足超谐性不等式，并在球域内得到定量对数凹性估计，丰富了分数阶椭圆方程基态的性质理论。
AI 中文摘要
设s∈(0,1)，我们证明：对任意开集Ω上的分数拉普拉斯算子(-Δ)^s的正基态u，只要其存在，就满足在Ω内有-Δu>λ_s(Ω)^(1/s)u，其中λ_s(Ω)为对应的第一特征值；在球B_R内，还得到定量对数凹性估计D²log u(x)≤D²log u(0)<-λ_s(B_R)^(1/s)/n · Id。
英文摘要
We prove that the positive ground state $u$ of the fractional Laplacian $(-Δ)^s$, $s\in (0,1)$, on an arbitrary open set $Ω$ satisfies $-Δu>λ_s(Ω)^{1/s}u$ in $Ω$. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for $\log u$, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of $ψ(-Δ)$ for complete Bernstein functions $ψ$.
