Two optimization problems in thermal insulation

We consider two optimization problems in thermal insulation. In both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The first problem deals with the case in which a given heat source is present. In the second case there are no heat sources, and the goal is to have the slowest decay of the temperature. In the first case, for a uniform heat source, the insulation is distributed uniformly about a ball, though for a square it is more concentrated on the middles of the sides. In the second case, even for a ball, when the amount of insulation falls below a certain threshold, it is better not to distribute it uniformly. We also provide numerical computations and a list of open questions