向量值自由边界问题平坦性的改进
Improvement of flatness for vector valued free boundary problems
摘要：
对于向量Bernoulli型自由边界问题，没有符号假设的组件，我们证明了平坦的自由边界意味着$C^{1,α}$正则性，众所周知的标量情况下\cite{AC,C2}。虽然在\cite{MTV2}中，通过使用标量问题的简化和自由边界的规则部分的NTA结构来最小化解，获得了相同的结果，但我们的结果直接使用了矢量问题的粘性方法，本着\cite{D}的精神。我们计划使用的方法在这里开发的矢量自由边界问题涉及一个分数拉普拉斯算子，在标量的情况下，在\cite{DR, DSS}治疗。
英文摘要：
For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,α}$ regularity, as well-known in the scalar case \cite{AC,C2}. While in \cite{MTV2} the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of \cite{D}. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in \cite{DR, DSS}.
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