This article is devoted to the study of two extremal problems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric fins and model this problem as the issue of (i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjoint operator having a compact inverse. We impose a pointwise lower bound on the radius of the fin, as well as a lateral surface constraint. Using particular perturbations and under a smallness assumption on the pointwise lower bound, one shows that the only solution is the cylinder in the first case whereas there is no solution in the second case. We moreover construct a maximizing sequence and provide the optimal value of the eigenvalue in this case. As a byproduct of this result, and to propose a remedy to the non-existence in the second case, we also investigate the well-posedness character of another optimal design problem set in a class enjoying good compactness properties.