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$$C^ \alpha{} ({\bar \Omega{}})$$ solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. (English) Zbl 0744.35016
This paper deals with a nonlinear degenerated highly coupled elliptic system which models a temperature dependent electrical resistor. The right-hand side here has quadratic growth in the gradient of one of the unknowns (the electrical potential). The existence of positive $$C^ \alpha(\overline\Omega)$$ solutions is proved for some $$0<\alpha<1$$ under realistic hypotheses on the data (boundary values, and electrical and thermal conductivity). To this end, the conductivities are truncated (as soon as a priori boundedness estimates are established for the solutions), which leads to a uniformly elliptic system. Now to obtain $$C^ \alpha(\overline\Omega)$$ estimates, $$L^{2,\mu}(\Omega)$$ estimates are used. Thus the Schauder fixed-point theorem applies, and the desired result follows.
Reviewer: C.Popa (Iaşi)

##### MSC:
 35J70 Degenerate elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35Q60 PDEs in connection with optics and electromagnetic theory 35J60 Nonlinear elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs
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