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.
Andrea Colesanti
. LOG-CONCAVITY OF THE FIRST DIRICHLET EIGENFUNCTION OF SOME ELLIPTIC DIFFERENTIAL OPERATORS AND CONVEXITY INEQUALITIES FOR THE RELEVANT EIGENVALUE[J]. Acta mathematica scientia, Series B, 2025
, 45(1)
: 143
-152
.
DOI: 10.1007/s10473-025-0111-0
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