2015/12/30 by Antonin Chambolle, Jimmy Lamboley, Chambolle, Antonin +5
Computer Science · Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #math.AP #math.OC
paper · pdf · doi:10.48550/arxiv.1512.08925
openalex publication_date 2015/12/30 · arxiv created 2016/04/15 · arxiv updated 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O§, 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.