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This paper settles the existence question for a rather general class ofconvex optimal design problems with a volume constraint. In low dimensions, weprove the existence of an optimal configuration for general convex minimizationproblems ruled by bounded measurable degenerate elliptic operators. Under amild continuity assumption on the medium, the free boundary is proven to enjoythe appropriate weak geometry and we establish the existence of an optimaldesign for general convex optimal design problems with volume constraints forall dimensions.

Optimal design problems in rough inhomogeneous media. Existence theory