Geodesic
On the existence of a generalized solution to a three-dimensional elliptic equation with radiation boundary condition
Applications of Mathematics, Tome 46 (2001) no. 4, pp. 241-250

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For a second order elliptic equation with a nonlinear radiation-type boundary condition on the surface of a three-dimensional domain, we prove existence of generalized solutions without explicit conditions (like $u\big |_\Gamma \in L_5(\Gamma )$) on the trace of solutions. In the boundary condition, we admit polynomial growth of any fixed degree in the unknown solution, and the heat exchange and emissivity coefficients may vary along the radiating surface. Our generalized solution is contained in a Sobolev space with an exponent $q$ which is greater than $9/4$ for the fourth power law.
For a second order elliptic equation with a nonlinear radiation-type boundary condition on the surface of a three-dimensional domain, we prove existence of generalized solutions without explicit conditions (like $u\big |_\Gamma \in L_5(\Gamma )$) on the trace of solutions. In the boundary condition, we admit polynomial growth of any fixed degree in the unknown solution, and the heat exchange and emissivity coefficients may vary along the radiating surface. Our generalized solution is contained in a Sobolev space with an exponent $q$ which is greater than $9/4$ for the fourth power law.
DOI : 10.1023/A:1013796024866
Classification : 35D05, 35J60, 35J65
Keywords: radiation boundary condition; generalized solution; existence 
Simon, László; Stoyan, Gisbert. On the existence of a generalized solution to a three-dimensional elliptic equation with radiation boundary condition. Applications of Mathematics, Tome 46 (2001) no. 4, pp. 241-250. doi: 10.1023/A:1013796024866
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     year = {2001},
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