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radiation boundary condition; generalized solution; existence
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.
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