Superharmonicity of the fractional ground state

Abstract

Let $s\in(0,1)$. We prove that the positive ground state $u$ of the fractional Laplacian $(-\Delta)^s$ on an arbitrary open set $\Omega$, whenever it exists, satisfies $-\Delta u>\lambda_s(\Omega)^{1/s}u$ in $\Omega$, where $\lambda_s(\Omega)$ denotes the corresponding first eigenvalue. In a ball $B_R$, we also obtain the quantitative log-concavity estimate $D^2\log u(x)\le D^2\log u(0) < -\frac{\lambda_s(B_R)^{1/s}}{n}\,\mathrm{Id}$.