I will discuss the problem of finding an optimal conductivity tensor field min-
imizing thermal compliance for a given balanced distribution of heat sources and
sinks. The heat sources are expressed as objects more general than Radon mea-
sures. The solution to the optimization problem is quite explicitly expressed
in terms of the data. The work is based on a foundation made by Bouchitte and
Buttazzo. I will also discuss the related problems concerning solving the heat
equation on lower dimensional structures in R^3 which are formed by a finite
union of one and two dimensional components.