Optimal design problems in rough inhomogeneous media. Existence theory

TL;DR

This paper proves the existence of optimal configurations in convex design problems within rough inhomogeneous media, addressing the challenge across various dimensions and under mild medium assumptions.

Contribution

It establishes the existence of solutions for a broad class of convex optimal design problems in inhomogeneous media, extending previous results to higher dimensions.

Findings

Existence of optimal configurations in low dimensions.

Existence of optimal designs in all dimensions under mild assumptions.

Weak geometric properties of free boundaries in the solutions.

Abstract

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 the 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 enjoy the appropriate weak geometry and we establish the existence of an optimal design for general convex optimal design problems with volume constraints for all dimensions.