An optimal design problem for a two-phase isolating material in the wall of a cavity

Abstract

We are interested in the optimal distribution of two isolating materials in the wall of a cavity in order to get the best isolation when the amount of the best isolating material is limited. We assume that the wall is of width ε>0 small. The problem can be formulated as the minimization of the first eigenvalue of a certain diffusion operator where the diffusion constant is of order one inside the cavity and of order ε in the wall. As it is usual in optimal design, the problem has not solution in general and therefore, it is necessary to work with a relaxed formulation. Passing to the limit when ε tends to zero we get an asymptotic model where the variable control is now in a Robin boundary condition instead of the diffusion operator. We also present some numerical simulations showing the behavior of the solutions.