Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. (English) Zbl 1035.49008

Bellido, José C.; Pedregal, Pablo

Discrete Contin. Dyn. Syst. 8, No. 4, 967-982 (2002).

Summary: We study relaxation for optimal design problems in conductivity in the two-dimensional situation. To this end, we reformulate the optimal design problem in an equivalent way as a genuine vector variational problem, and then analyze relaxation of this new variational problem. Our main achievement is to explicitly compute the quasiconvexification of the involved density in this problem for some interesting cases. We think the method given here could be generalized to compute quasiconvex envelopes in other situations. We restrict attention to the two-dimensional case

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
49J10	Existence theories for free problems in two or more independent variables
78A30	Electro- and magnetostatics
78M40	Homogenization in optics and electromagnetic theory

Keywords:

dependence on the gradient of the state; relaxation; optimal design; conductivity; vector variational problem; quasiconvexification