Radiative effects on the thermoelectric problems

There are two main directions in this paper. One is to find sufficient conditions to ensure the existence of weak solutions to thermoelectric problems. At the steady-state, these problems consist by a coupled system of elliptic equations of the divergence form, commonly accomplished with nonlinear radiation-type conditions on at least on a nonempty part of the boundary of a $C^1$ domain. The model under study takes the thermoelectric Peltier and Seebeck effects into account, whose describe the Joule-Thomson effect. The proof method makes recourse of a fixed point argument. To this end, well-determined estimates are our main concern. The paper is in the second direction for the derivation of explicit $W^{1,p}$-estimates $(p>2)$ for solutions of nonlinear radiation-type problems, where the leading coefficient is assumed to be a discontinuous function on the space variable. In particular, the behavior of the leading coefficient is conveniently explicit on the estimate of any solution. This regularity result is sufficiently general to contribute to other problems, in which the dependence on the values of the involved constants is essential, instead of the problem under study only.