Problems on the loss of heat: herd instinct versus individual feelings

Several problems are discussed concerning steady-state distribution of heat in domains in R 3 \mathbb {R}^3 that are complementary to a finite number of balls. The study of these problems was initiated by M. L. Glasser in 1977. Then, in 1978, M. L. Glasser and S. G. Davison presented numerical evidence that the heat flux from two equal balls in R 3 \mathbb {R}^3 decreases when the balls move closer to each other. Those authors interpreted this result in terms of the behavioral habits of sleeping armadillos, the closer animals to each other, the less heat they lose. Much later, in 2003, A. Eremenko proved this monotonicity property rigorously and suggested new questions on the heat fluxes. The goal of this paper is to survey recent developments in this area, provide answers to some open questions, and draw attention to several challenging open problems concerning heat fluxes from configurations consisting of n ≥ 2 n\ge 2 balls in R 3 \mathbb {R}^3 .