Power-logconcavity of the Laplacian ground state

Abstract

Let $u$ be the first Dirichlet Laplacian eigenfunction of a bounded convex set $\Omega$ in ${\mathbb{R}}^n$. We strengthen the classical result by Brascamp-Lieb which asserts that $u$ is logconcave in $\Omega$: we prove that, if $u$ is normalized so that its $L^{\infty }$-norm does not exceed a threshold $\overline{\kappa }(\Omega )<1$ depending explicitly on the diameter of the domain and on its principal frequency, the function $-(-\log u{)}^{1/2}$ is concave in $\Omega$.

14 pages, 1 figure