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

We study the infimum of functionals of the form $int_Omega Mnabla ucdotnabla u$ among all emph{convex functions $uin H^1_0(Omega)$ such that $int_Omega abs{nabla u^2 =1$. ($Omega$~is a convex open subset of $R^N$, and $M$ is a given symmetric $Ntimes N$ 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 $M$.