What is the optimal shape of a fin for stationary heat conduction?

This article is concerned with the shape of small devices used to control the heat flowing between a solid and a fluid phase, usually called fin. The temperature along a fin in stationary regime is modeled by a one-dimensional Sturm-Liouville equation whose coefficients strongly depend on its geometrical features. We are interested in the following issue: is there any optimal shape maximizing the heat flux at the inlet of the fin? Two relevant constraints are examined, by imposing either its volume or its perimeter, and nonexistence results are proved for both problems. Furthermore, using a kind of rearrangement argument, we explicitly compute the optimal values and construct maximizing sequences. We show in particular that the optimal heat flux at the inlet is infinite in the first case and finite in the second one. Finally, we provide several extensions of these results for more general models of heat conduction, as well as several numerical illustrations.