Circular symmetrization and extremal Robin conditions

We study the average temperature in a homogeneous disk subject to uniform heating in its interior and Newton's law of cooling, hv + partial derivative v/partial derivative n = 0, on its boundary. More precisely, among those h taking values in a prescribed interval, and of prescribed mean, we identify the minimizer and a maximizer of the average temperature. The latter characterization makes use of circular symmetrization. We include an elementary proof of the fact that such symmetrization does not increase the Dirichlet energy of any H-1 function on the disk.