  We consider the problem of optimally insulating a given domain $\Omega$ of
${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where
the optimal thickness of the insulator is obtained as the boundary trace of the
solution. We deal with two different criteria of optimization: the first one
consists in the minimization of the total energy of the system, while the
second one involves the first eigenvalue of the related differential operator.
Surprisingly, the second optimization problem presents a symmetry breaking in
the sense that for a ball the optimal thickness is nonsymmetric when the total
amount of insulator is small enough. In the last section we discuss the shape
optimization problem which is obtained letting $\Omega$ to vary too.
