Optimal design problems in rough inhomogeneous media: existence theory

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to have an appropriate weak geometry and we establish existence of an optimal design in any dimension.