Hölder continuous solutions of an obstacle thermistor problem (Q1884253)

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.