Level set method via positive parts for optimal design problems of two-material thermal conductors

This paper is concerned with optimal design problems for two-material thermal conductors via level set methods based on (doubly) nonlinear diffusion equations. In level set methods with material representations via characteristic functions, gradient descent methods cannot be applied directly in terms of the differentiability of objective functionals with respect to level set functions, and therefore, appropriate sensitivities need to be constructed. This paper proposes a formulation via the positive parts of level set functions to avoid heuristic derivation of sensitivities and to apply (generalized gradient) descent methods. In particular, some perturbation term, such as a perimeter constraint, is involved in the formulation, and then an existence theorem for minimizers will be proved. Furthermore, convergence of objective functionals for minimizers with respect to a parameter of the perturbation term will also be discussed. In this paper, by deriving so-called weighted sensitivities, two-phase domains are numerically constructed as candidates for optimal configurations to approximate minimum values for classical design problems in two-dimensional cases.