The minimum of quadratic functionals of the gradient on the set of convex functions

Abstract

We study the infimum of functionals of the form $\int_\Omega M\nabla u\cdot\nabla u$ among all convex functions $u\in H^1_0(\Omega)$ such that $\int_\Omega |\nabla u|^2 =1$ . ( $\Omega$ is a convex open subset of ${\mathbb R}^N$ , and M is a given symmetric $N\times N$ matrix.) We prove that this infimum is the smallest eigenvalue of M if $\Omega$ is $C^1$ . Otherwise the picture is more complicated. We also study the case of an x-dependent matrix M.