Optimal design of a bar with an attached mass for maximizing the heat transfer

We maximize, with respect to the cross sectional area, the rate of heat transfer through a bar of given mass. The bar serves as an extended surface to enhance the heat transfer surface of a larger heated known mass to which the bar is attached. In this paper we neglect heat transfer from the sides of the bar and consider only conduction through its length. The rate of cooling is defined by the first eigenvalue of the corresponding Sturm-Liouville problem. We establish existence of an optimal design via rearrangement techniques. The necessary conditions of optimality admit a unique optimal design. We compare the rate of heat transfer for that bar with the rate for the bar of the same mass but of a constant cross-section area.