Non-local Problem Arising from Heat Radiation on Non-convex Surfaces Non-local Problem Arising from Heat Radiation on Non-convex Surfaces

We consider both stationary and time dependent heat equation for a non-convex body or a collection of disjoint conductive bodies with Stefan-Boltzmann radiation condition on the surface. The main virtue of the resulting problem is non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is non-linear and in the general case also non-coercive. We show that the non-local boundary value problem has a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. This result is then applied to prove the existence under some hypotheses that guarantee the existence of sub-and supersolutions. Some special cases where the problem is coercive are also discussed. Finally, the analysis is extended to cases with non-linear material properties.