Calculus of Variations and Geometric Measure Theory
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R. De Arcangelis

Homogenization of Dirichlet Minimum Problems with Conductor Type Periodically Distributed Constraints

created on 16 Dec 2002
modified on 20 Dec 2002


Published Paper

Inserted: 16 dec 2002
Last Updated: 20 dec 2002

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

Studies in Mathematics and its Applications 31, North-Holland


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.

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