Regularity for the optimal compliance problem with length penalization

TL;DR

This paper establishes regularity properties of optimal sets minimizing compliance plus length in a planar domain, showing they are composed of smooth curves meeting at 120 degrees without loops, advancing understanding of elliptic PDE shape optimization.

Contribution

It provides new regularity results and geometric structure theorems for minimizers in a length-penalized compliance optimization problem, connecting PDE and geometric measure theory.

Findings

Minimizers are finite unions of smooth curves

Curves meet only in triples at 120 degrees

Minimizers contain no loops and touch boundary tangentially

Abstract

We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O\S, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching the boundary of the domain only tangentially. Several new technical tools together with the classical ones are developed for this purpose.