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Hölder continuous solutions of an obstacle thermistor problem. (English) Zbl 1099.35183
The authors obtain a regularity result for the solution of an obstacle thermistor problem. The nonlinear problem studied has nonlocal boundary condition and the existence of a Hölder continuous $$C^{\alpha,\alpha/2}$$ solution represents the center focus of the article. The proof is based on a penalized method introduced by the authors in a previous article and also a technique that involves the Leray-Schauder degree theory along with results for Campanato spaces. Using a decomposition method for transforming the equation satisfied by the electric potential into two elliptic problems, it is proved, first, the existence of $$C^{\alpha,\alpha/2}$$ solutions for the penalized equations and then, a solution of the original obstacle system is obtained by passing to the limit of some subsequence of the penalized solutions.

##### MSC:
 35R45 Partial differential inequalities and systems of partial differential inequalities 49J40 Variational inequalities 74F15 Electromagnetic effects in solid mechanics
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