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 is, here, 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 (monotonicty formula, flatness improving estimates, blow-up limits), but adapted to the specific problem of min-max type studied here, which constitutes a notable difference with the classical setting and may be useful also for similar other problems. ∗CMAP, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France †Université Paris-Dauphine, PSL Research University, UMR 7534, CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France ‡Université Paris Diderot LJLL CNRS UMR 7598 Paris 7 75205 Paris Cedex 13, France §St.Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, Fontanka 27, 191023 St.Petersburg,Russia and Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St.Petersburg, Russia and ITMO University, Russia ¶This work was partially supported by the projects ANR-12-BS01-0007 OPTIFORM and ANR-12-BS01-001401 GEOMETRYA financed by the French Agence Nationale de la Recherche (ANR). The authors acknowledge the support of the project MACRO (Modèles d’Approximation Continue de Réseaux Optimaux), funded by the Programme Gaspard Monge pour l’Optimisation of EDF and the Fondation Mathématiques Jacques Hadamard. The fourth author also acknowledges the support of the St.Petersburg State University grants #6.38.670.2013 and #6.38.223.2014, the Russian government grant NSh-1771.2014.1, the RFBR grant #14-01-00534 and the project 2010A2TFX2 “Calcolo delle variazioni” of the Italian Ministry of Research.