Regularity for the optimal compliance problem with length penalization
We study the regularity and topological structure of a compact connected set $S$ minimizing the “compliance" functional with a length penalization. The compliance here is the work of the force applied to a membrane which is attached along the set $S$. This shape optimization problem, which can be interpreted as that of finding the best location for attaching a membrane subject to a given external force, can be seen as an elliptic PDE version of the minimal average distance problem. We prove that minimizers in the given region consist of a finite number of smooth curves which meet only at triple points with angles of 120 degrees, contain no loops, and possibly touch the boundary of the region only tangentially. The proof uses, among other ingredients, some tools from the theory of free discontinuity problems (monotonicity formula, flatness improving estimates, blow-up limits), but adapted to the specific problem of min-max type studied here, which constitutes a significant difference with the classical setting and may be useful also for similar other problems.