Regularity for the Optimal Compliance Problem with Length Penalization

Abstract

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.

A selective review on Mumford–Shah minimizers

“…In conclusion we have obtained that Up to now, everything can be rigorously justified (see [CLLS,BLS15]). But to go further, we will be more optimistic, and try to write Φ as the gradient of something.…”

“…In [BS07], the following free discontinuity problem is introduced, also studied in [Til12,CLLS]. It is commonly called the "Glue-problem" by Buttazzo, and has apparently nothing to do with the Mumford-Shah problem.…”

Regularity for the planar optimalp-compliance problem

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In this paper we prove a partial C 1,α regularity result in dimension N = 2 for the optimal p-compliance problem, extending for p = 2 some of the results obtained by Chambolle et al (2017). Because of the lack of good monotonicity estimates for the p-energy when p = 2, we employ an alternative technique based on a compactness argument leading to a p-energy decay at any flat point.

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“…in the weak sense, which means that Ω |∇u Σ | p−2 ∇u Σ ∇ϕ dx = Ω f ϕ dx (1.3) for all ϕ ∈ W 1,p 0 (Ω \ Σ). Following [11], we can interpret Ω as a membrane which is attached along Σ ∪∂ Ω to some fixed base (where Σ can be interpreted as a "glue line") and subjected to a given force f . Then u Σ is the displacement of the membrane.…”

Optimal one-dimensional structures for the principal eigenvalue of two-dimensional domains

“…Problem 2. For the optimal set S of problem (2.5) several necessary conditions of optimality merit to be investigated, for instance we list the following ones, that look similar to other problems studied in the fields of optimal transport and of structural mechanics (see [6], [8], [9]). Problem 3.…”