  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 \textsl{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
surface, and analytical nonexistence results are proved for both problems.
Furthermore, using specific perturbations, 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.
