C a Ω¯solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem

Under realistic assumptions on the electrical and thermal conductivities, the existence is proven, for some $0 < \alpha < 1$, of positive $C^\alpha (\bar \Omega )$ solutions for a system of degenerate elliptic equations which model a thermistor. A priori bounds are established for the solutions, and then the conductivities are truncated, so that a uniformly elliptic system is obtained. Next, $L^{2,\mu } (\Omega )$ estimates are used to obtain $C^\alpha (\bar \Omega )$ estimates. Finally, the desired results follow from fixed-point