Deterministic homogenization of integral functionals with convex integrands
type: Item
instance of: scholarly article
title: Deterministic homogenization of integral functionals with convex integrands
zbMATH Open document ID: 1207.35046
author: Gabriel Nguetseng
author: Hubert Nnang
author: Jean Louis Woukeng
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2010-12-03T00:00:00Z
review text: The authors develop a framework adapted to the homogenization of problems involving integral functionals and based on the concept of \(H\)-algebra. The generic problem is written as \(\min \{F_\varepsilon(v):v\in W_0^{1,p}(\Omega ;\mathbb R^n)\}\) where the functional \(F_\varepsilon\) is defined on \(W_{0}^{1,p}(\Omega ;\mathbb R^{n})\) by \(F_{\varepsilon }(v)=\int_{\Omega }f(x,x/\varepsilon ,Dv)\,dx\). Here \(\Omega \) is a bounded and open subset of \(\mathbb R^{N}\), and \(f\) is a function from \(\mathbb R^{N}\times\mathbb R^{N}\times\mathbb R^{nN}\) to \([0,+\infty )\) which satisfies some hypotheses among which are smoothness, measurability and a \(p\) -growth conditions. The main tools of the study are the definitions of homogenization algebras and of a weak \(\Sigma\)-convergence for sequences of functions in \(L^{p}(\Omega )\). The main result of the paper establishes the structure of the limit integral functional which involves an appropriate \(H\)-algebra \(A\) which is supposed to satisfy some ``proper'' further hypothesis. This main result also proves the weak convergence of the sequence \( (u)_{\varepsilon }\) in \(W_{0}^{1,p}(\Omega ;\mathbb R^{n})\) and the weak convergence to \((\frac{\partial u}{\partial x_{i}})_{\varepsilon }\) in \( L^{p}(\Omega )^{n}\). The paper ends with some applications of the general result to some concrete situations. The authors start with a periodic case following with almost periodic cases.
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 5822925
zbMATH Keywords: homogenization algebra
zbMATH Keywords: \(\Sigma\)-convergence
zbMATH Keywords: periodic case
zbMATH Keywords: almost periodic cases
MaRDI profile type: Publication
full work available at URL: s00030-010-0080-3
OpenAlex ID: W2063477643
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DOI: 10.1007/S00030-010-0080-3
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description: scientific article; zbMATH DE number 5822925