Calculus of Variations and Homogenization
François Murat, Luc C. Tartar · Birkhäuser Boston eBooks · 1997
In this paper we study a class of problems which includes the following example. Let Ω be an open subset of IR N 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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.