Symmetrization and optimal control for elliptic equations

Abstract:

We consider an optimal control problem where $u(x)$ satisfies $- \operatorname {div}(H(x)\nabla u) = 1$ in $\Omega$ and $H(x)$ is a control. We introduce the functional ${J_\Omega }(H) = {|\Omega |^{ - 1}}\int \limits _\Omega {u(x)} dx$ and show using a symmetrization argument that if the distribution function of $H$ is fixed, then ${J_\Omega }(H)$ is largest when $\Omega$ is a ball and $H$ is radial and decreasing on radii.