A free discontinuity approach to optimal profiles in Stokes flows

Abstract

In this paper we study obstacles immersed in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension $1$. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation ($\mathrm{SBD}$). In dimension two, we prove that the solution is classical.