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A nonlocal problem arising from heat radiation on non-convex surfaces

Published online by Cambridge University Press:  01 August 1997

T. TIIHONEN
Affiliation:
University of Jyväskylä, Laboratory of Scientific Computing, PO Box 35, FIN–40351 Jyväskylä, Finland

Abstract

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.

Type
Research Article
Copyright
© 1997 Cambridge University Press

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