TY - JOUR

T1 - Regularity for the optimal compliance problem with length penalization

AU - Chambolle, Antonin

AU - Lamboley, Jimmy

AU - Lemenant, Antoine

AU - Stepanov, Eugene

N1 - Funding Information: The work of the second author was partially supported by the project ANR-12-BS01-0007 OPTIFORM. The work of the second and third authors was supported by ANR-12-BS01-0014-01 GEOMETRYA financed by the French Agence Nationale de la Recherche (ANR), the project MACRO (Mod?les d'Approximation Continue de R?seaux Optimaux), funded by the Programme Gaspard Monge pour l'Optimisation of EDF, and the Fondation Math?matiques Jacques Hadamard. The fourth author also acknowledges the support of the Russian government grant 074-U01, of the Ministry of Education and Science of Russian Federation project 14.Z50.31.0031, and of the RFBR grant 17-01-00678. Publisher Copyright: © 2017 Society for Industrial and Applied Mathematics. Copyright: Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Compliance

KW - Mumford-Shah

KW - Regularity theory

KW - Shape optimization

UR - http://www.scopus.com/inward/record.url?scp=85018784010&partnerID=8YFLogxK

U2 - 10.1137/16M1070578

DO - 10.1137/16M1070578

M3 - Article

AN - SCOPUS:85018784010

VL - 49

SP - 1166

EP - 1224

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -