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

Regularity for the optimal compliance problem with length penalization