Calculus of Variations and Geometric Measure Theory

R. De Arcangelis

Homogenization of Dirichlet Minimum Problems with Conductor Type Periodically Distributed Constraints

created on 16 Dec 2002
modified on 20 Dec 2002

[BibTeX]

Published Paper

Inserted: 16 dec 2002
Last Updated: 20 dec 2002

Journal: "Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Volume XIV, Doina Cioranescu and Jacques-Louis Lions Editors"
Volume: 31
Pages: 243-271
Year: 2002
Notes:

Studies in Mathematics and its Applications 31, North-Holland


Abstract:

The homogenization of Dirichlet minimum problems is studied for integral functionals of the type $$u\in u{0,h}+W{1,\infty}0\mapsto\int\Omega f(hx,\nabla u)dx,$$ where $\Omega$ is a smooth bounded open set, $\{u_{0,h}\}$ is a converging sequence in $L^1(\Omega)$, and when the admissible functions are subject to a constraint of the type $$u\hbox{ is constant in }\Omega\cap{1\over h}S\hbox{ for every }S\in{\cal C},$$ where $\cal C$ is a $]0,1[^n$-periodic collection of subsets of $*R*^n$, and the constant value of $u$ is not fixed a priori.

An homogenization formula is proved, and the homogenization process is proved in both the settings of Sobolev and $BV$ spaces.

Some examples are also discussed.