A \(\Gamma\)-convergence result for optimal design problems
type: Item
instance of: scholarly article
title: A \(\Gamma\)-convergence result for optimal design problems
zbMATH Open document ID: 1501.74062
author: Hamdi Zorgati
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2022-10-12T00:00:00Z
review text: The author considers optimal design models of a two-phase elastic mixture described by functionals \begin{align*} &I_p(\chi,u)=\left(\int_\Omega\left(\chi W_1(\nabla u)^p+(1-\chi)W_2(\nabla u)^p\right)\,dx\right)^{\frac 1p},\\ & \text{if}\ (\chi,u)\in L^\infty(\Omega;\{0,1\})\times W^{1,p}(\Omega;\mathbb{R}^m),\ \text{or} +\infty\ \text {otherwise}, \end{align*} where \(p\equiv \{p_n\},\ 1<p_n\to +\infty\) and \(W_i^p,\ i=1,2\), models the energy density of the \(i\)th phase. The functionals represent the elastic energy of the solid occupying the domain \(\Omega\) and undergoing the deformation \(u\), while \(\chi\) represents the characteristic functions of the first phase of stiffness or electric resistivity. The main result of the paper is the \(\Gamma\)-convergence of functionals \((I_p)_{p>p_0}\) as \(p\) goes to \(+\infty\) with respect to the \(L^\infty(\Omega;\{0,1\})\) weak-star and \(W^{1,p_0}(\Omega;\mathbb{R}^m)\) weak topology.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 35Q74
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7600083
zbMATH Keywords: energy functional
zbMATH Keywords: energy minimization
zbMATH Keywords: optimal material distribution
zbMATH Keywords: two-phase elastic mixture
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: cross-quasiconvexity
MaRDI profile type: Publication
full work available at URL: crmath.375
OpenAlex ID: W4300978909
cites work: Power-Law Approximation under Differential Constraints
cites work: On the lower semicontinuity of supremal functional under differential constraints
cites work: Relaxation for optimal design problems with non-standard growth
cites work: $\Gamma$-Convergence of Power-Law Functionals, Variational Principles in $L^{\infty},$ and Applications
cites work: 3D-2D asymptotic analysis for inhomogeneous thin films
cites work: 3D-2D dimensional reduction for a nonlinear optimal design problem with perimeter penalization
cites work: Γ-convergence and absolute minimizers for supremal functionals
cites work: An introduction to \(\Gamma\)-convergence
cites work: Q4075901
cites work: 3D-2D asymptotic analysis of an optimal design problem for thin films
cites work: Dielectric breakdown: optimal bounds
cites work: A Note on Optimal Design for Thin Structures in the Orlicz–Sobolev Setting
cites work: Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results
cites work: Optimal design of fractured media with prescribed macroscopic strain
DOI: 10.5802/CRMATH.375
description: scientific article; zbMATH DE number 7600083