  This paper is concerned with configurations of two-material thermal
conductors that minimize the Dirichlet energy for steady-state diffusion
equations with nonlinear boundary conditions described mainly by maximal
monotone operators. To find such configurations, a homogenization theorem will
be proved and applied to an existence theorem for minimizers of a relaxation
problem whose minimum value is equivalent to an original design problem. As a
typical example of nonlinear boundary conditions, thermal radiation boundary
conditions will be the focus, and then the Fr\'echet derivative of the
Dirichlet energy will be derived, which is used to estimate the minimum value.
Since optimal configurations of the relaxation problem involve the so-called
grayscale domains that do not make sense in general, a perimeter constraint
problem via the positive part of the level set function will be introduced as
an approximation problem to avoid such domains, and moreover, the existence
theorem for minimizers of the perimeter constraint problem will be proved. In
particular, it will also be proved that the limit of minimizers for the
approximation problem becomes that of the relaxation problem in a specific
case, and then candidates for minimizers of the approximation problem will be
constructed by employing time-discrete versions of nonlinear diffusion
equations. In this paper, it will be shown that optimized configurations deeply
depend on force terms as a characteristic of nonlinear problems and will also
be applied to real physical problems.
