Power-logconcavity of the Laplacian ground state

TL;DR

This paper proves a new power-logconcavity property of the first Dirichlet Laplacian eigenfunction in convex domains, extending classical logconcavity results under specific normalization conditions.

Contribution

It establishes that, under certain normalization, the transformed eigenfunction exhibits concavity, strengthening the classical logconcavity result by Brascamp-Lieb.

Findings

The function $- ( - ext{log} u )^{1/2}$ is concave in $ ext{Omega}$ under normalization.

The result depends explicitly on the domain's diameter and principal frequency.

It provides a new geometric insight into eigenfunctions of the Laplacian.

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 $\Omega$.