Approximation in an optimal design problem governed by the heat equation

This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In general, such optimal designs are not guaranteed to exist, and geometric constraints such as the perimeter are required. As an approximation of the problem with an additional perimeter constraint, a material representation based on a level set function, together with a perturbation of the Dirichlet energy, is employed. It is then shown that optimal level set functions exist for the perturbation problem, and the corresponding minimum value converges to that of the elliptic case, thereby elucidating the long-time behavior. Furthermore, two-material domains satisfying this property are also constructed via the nonlinear diffusion-based level set method. In particular, the asymptotic behavior with respect to the perturbation parameter is clarified, and the validity of the approximation is established.