Local extrema for energy in two-phase optimal design problems

Abstract

We consider conductivity optimal design problems for two isotropic phases with prescribed amounts, optimizing
the energy functional.

We analyze optimality conditions obtained by shape calculus, in the spherically symmetric case
for simple radial designs, such that the interface between two given isotropic phases consists of a single sphere.
Every such design satisfies the first-order optimality condition.
By using classical Fourier analysis techniques we are able
to express and analyze the second-order optimality conditions. This implies that, for any outer heat source, considered simple designs cannot
give a local minima of the energy functional.

Two examples are presented, showing that the presented approach gives a complete answer whether a considered critical design is a saddle point,
or the Lagrangian satisfies the negative coercivity condition in H$^{\frac12}$ norm of normal perturbation on the interface.
In the latter case, we numerically confirm the appearance of local maxima, different from the global one.