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Zbl 1151.35319
Druet, Pierre-Étienne
Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^{p} (p \geqslant 1)$. (English)
[J] Math. Methods Appl. Sci. 32, No. 2, 135-166 (2009). ISSN 0170-4214; ISSN 1099-1476/e

Summary: Accurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than $L^{1}$. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and $L^p$ right-hand side, with $p \geqslant 1$ being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the $L^p$ norm of the heat sources and to prove the existence of weak solutions.

MSC 2000:: *35D05 Existence of generalized solutions of PDE
35J60 Nonlinear elliptic equations

Keywords: existence of generalized solutions; nonlinear PDE of elliptic type; radiative heat transfer; nonlocal boundary condition; right-hand side $L^{1}$

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