We consider viscosity solutions to a one-phase free boundary problem for the $p\left(x\right)$-Laplacian with non-zero right hand side. We apply the tools developed in De Silva (2011) to prove that flat free boundaries are $C^{1,\alpha }$. Moreover, we obtain some new results for the operator under consideration that are of independent interest.

中文翻译：

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平面自由边界的正则性 $磷\left(X\right)$- 右手边的拉普拉斯问题

我们考虑单相自由边界问题的粘度解 $磷\left(X\right)$- 右手边非零的拉普拉斯算子。我们应用 De Silva (2011) 开发的工具来证明平坦的自由边界是$C^{1,\alpha }$. 此外，我们为所考虑的运营商获得了一些具有独立利益的新结果。