Year: 2026
Type: preprint
Abstract: 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 $ψ$.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: National Science Foundation, Istituto Nazionale di Alta Matematica "Francesco Severi", Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni, Staatssekretariat für Bildung, Forschung und Innovation, Agencia Estatal de Investigación
OpenAlex ID: W7202394015