Multiphase Shape Optimization Problems

Abstract

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min{g ((F 1 For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., F i = λ k i .

Lipschitz Regularity of the Eigenfunctions on Optimal Domains

“…In fact, for the functions v n = v n − u ∞ , we have that 10) and v n → 0 uniformly on B R/2 . By Remark 2.3, we have that ∇ v n L ∞ (B R/4 ) → 0 and so, v n → u ∞ in H 1 (B R/4 ) and the same holds for u n .…”

Minimal Dirichlet Energy Partitions for Graphs

“…Existence of optimal partitions for (2.4) in the class of quasi-open sets was proved in [6]. Subsequently, several papers have investigated (2.4) and similar problems, focusing on the regularity of partitions, properties of optimal partitions, the asymptotic behavior of optimal partitions as k → ∞, and computational methods [8,2,3,22,21,23,34,7]. In particular, the relaxation of the eigenvalue partitioning problem proposed here is analogous to a relaxation of (2.4) proposed in [3].…”

Existence and Regularity Results for Some Shape Optimization Problems

“…The results from Chapter 4, concerning the energy subsolutions, are from the recent paper [29]. Our main technical results, which are essential in the study of the qualitative properties of families of disjoint subsolutions (which naturally appear in the study of multiphase shape optimization problems) are a density estimate and a three-phase monotonicity theorem in the spirit of the two-phase formula by Caffarelli, Jerison and Kënig.…”

“…In all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation (shortly, PDE). In [29], we investigated this notion obtaining a density estimate, which we used to prove a regularity result for the optimal set for the second eigenvalue λ 2 in a box, and a three-phase monotonicity formula of Cafarelli-Jerison-Kënig type, which allowed us to exclude the presence of triple points in some optimal partition problems. The simplest state functions are provided by solutions of the equations − w = 1 and − u = λu, which usually represent the torsional rigidity and the oscillation modes of a given object.…”