Calculus of Variations and Homogenization

  • François Murat
  • Luc Tartar
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 31)


In this paper we study a class of problems which includes the following example. Let Ω be an open subset of IRN and let ω be a measurable subset of Ω with given measure γ’. Let a(x)= α on ω and β on Ω\ω, and define u by −div (agradu) = 1 in Ω and u = 0 on the boundary. We want to find an ω which maximizes \( \int {_{\Omega }} u(x) \) u (x) dx among all the measurable sets ω with given measure γ’.

This optimal design problem has in general no solution. Indeed if there exists a smooth solution ω and if Ω is simply connected, then Ω is a ball. In general a maximizing sequence does not converge to a set ω, but to a “generalized domain”, i.e. a more and more intricate mixture of materials α and β (homogenization phenomenon). Since we know the characterization of all the materials obtained in this way, i.e. the optimal bounds for homogenized materials, we can write a relaxed problem of the initial one. This relaxed problem has a solution, but the parameter is now a generalized (homogenized) domain.

We then study the necessary conditions for a generalized domain to be an optimal solution. This leads to some characterizations and to several geometrical remarks about the solutions of the relaxed problem and those of the initial one, if they exist.


Characteristic Function Homogenize Region Generalize Domain Optimal Design Problem Compact Topological Space 
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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • François Murat
  • Luc Tartar

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