Abstract.We study the well-posechiess of a class of models describing heat transfer by conduction and radiation.For that purpose we propose an abstract mathematical framework that allows us to prove existence, uniqueness and the comparison principle for the weak solution with minimal or almost minimal a priori assumptions for the data. The theory covers different types of grey materials, that is, both semitransparent and opaque bodies as well as isotropic or nonisotropic scattering/reflection provided that the material properties do not depend on the wavelength of the radiation. To demonstrate the use of the abstract theory we consider in detail two examples, heat transfer between opaque bodies with diffuse-grey surfaces and a model with semitransparent material and specularly reflecting surfaces.

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We consider the stationary heat equation for a non‐convex body with Stefan–Boltzmann radiation condition on the surface. The main virtue of the resulting problem is non‐locality of the boundary condition. Moreover, the problem is non‐linear and in the general case also non‐coercive and non‐monotone. We show that the 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. In the two dimensional case or when a part of the radiation can escape the system we obtain coercivity and stronger existence result. © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.

In this paper we consider numerical methods for stationary free boundary problems. We start by analysing systematically di erent shape optimization formulations of a model problem and show how the optimality conditions relate to construction of trial type methods. Shape sensitivity analysis of the free boundary leads also to the so-called total linearization method which combines the good properties of Newton method and trial methods, i.e. fast convergence and relative simplicity of implementation. Detailed implementation for a model problem together with numerical tests is presented.

We consider both stationary and time-dependent heat equations for a non-convex body or a collection of disjoint conducting bodies with Stefan-Boltzmann radiation conditions on the surface. The main novelty of the resulting problem is the non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is nonlinear and in the general case also non-coercive. We show that the non-local boundary value problem admits 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 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 nonlinear material properties.