To $1/2$-logconcavity and beyond: Geometric properties of Dirichlet eigenfunctions

Abstract

We prove that, on a bounded open convex domain $Ω\subset\mathbb{R}^n$, the first Dirichlet eigenfunction of the Laplacian or the Ornstein--Uhlenbeck operator is $α$-logconcave for every $α\in(0,1/2]$. This extends the recent $1/2$-logconcavity theorem of Crasta--Fragalà for the Laplacian to the weighted Gaussian setting and, simultaneously, to a broader range of exponents. More precisely, if $u$ denotes the first eigenfunction normalized by $\|u\|_\infty=1$, then for every $α\in(0,1/2]$, the function $-\bigl(-\log(κu(x))\bigr)^α$ is concave in $Ω$ provided the scaling parameter $κ$ lies below an explicit threshold $κ_α(Ω)\in(0,1)$, which depends on the first Dirichlet eigenvalue and on the diameter of~$Ω$. For the Ornstein--Uhlenbeck operator, $κ_α(Ω)$ also depends on the distance between $Ω$ and the origin. Moreover, we establish a local counterpart: for every $κ\in(0,1)$, the function $\bigl(-\log(κu)\bigr)^α$ is convex on a convex neighborhood $Ω_κ$ of the unique maximum point of~$u$. We also provide counterexamples showing that unscaled $1/2$-logconcavity may fail for the first Dirichlet eigenfunction of a Schrödinger operator with a smooth convex potential, and for the first Dirichlet eigenfunction of a weighted Laplacian associated with an affine log-concave weight.