Symmetrization and optimal control for elliptic equations

Abstract

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