关键词: eigenvalue, log-concavity, elliptic operator, Brunn-Minkowski inequality, convex body

Abstract: Given an open bounded subset $\Omega$ of $\mathbb{R}^{n}$ we consider the eigenvalue problem $\begin{equation*} \left\{ \begin{array}{lll} \Delta u-\langle\nabla u,\nabla V\rangle=-\lambda_V u,\quad u>0\quad&\mbox{in $\Omega$,}\\ u=0\quad&\mbox{on $\partial\Omega$,} \end{array} \right. \end{equation*}$ where $V$ is a given function defined in $\Omega$ and $\lambda_V$ is the relevant eigenvalue. We determine sufficient conditions on $V$ such that if $\Omega$ is convex, the solution $u$ is log-concave. We also determine sufficient conditions ensuring that $\lambda_V$, as a function of the set $\Omega$, verifies a convexity inequality with respect to the Minkowski addition of sets.

Key words: eigenvalue, log-concavity, elliptic operator, Brunn-Minkowski inequality, convex body