A temperature control problem

Abstract

In an open bounded region ofn-space occupied by a homogeneous and isotropic medium, we control the temperature through the boundary. The normal derivative of the temperature (which measures the appropriate heat flux) is restricted to be nonnegative. This gives rise to a free boundary in space-time separating the areas of positive and zero heat flux. Under a natural monotonicity condition, the free boundary is the graph of a function of space. This function is shown to be locally Lipschitz. Moreover forn=2the time derivative of the temperature is proven to be continuous across the free boundary.