Existence of solution of a coupled problem arising in the thermoelectricalsimulation of electrodes

In this paper we prove the existence of a solution to a system of partial d ifferential equations arising from the thermoelectrical modeling of electro des for electric furnaces. It consists of Maxwell equations coupled with th e heat transfer equation through the Joule effect and the fact that thermal conductivity and electrical resistivity depend on temperature. The problem is formulated in cylindrical coordinates to take advantage of its axisymme try. The result is shown by introducing a regularized problem and using Schauder 's fixed point theorem. Passing to the limit requires a priori estimates in weighted Sobolev spaces for an elliptic problem involving a right-hand sid e that is only integrable.