AI 中文总结
本文针对无凸性假设的完全非线性Alt-Phillips自由边界问题，通过部分速端变换转化方程，证明Harnack不等式与线性化方程Schauder估计，最终得到平坦自由边界的光滑性结论，相关估计也具独立研究意义。
AI 中文摘要
本文研究γ∈(-1/3,1)时的完全非线性Alt-Phillips自由边界问题F(D²u)=u^γχ_{{u>0}}，不对F施加任何凸性假设。我们证明了平坦自由边界是光滑的，该结果即使在γ=0的完全非线性障碍问题情形下也属全新。我们的方法基于部分hodograph（速端）变换，该变换将自由边界转化为固定边界，并导出一个完全非线性退化方程。针对该方程，我们证明了Harnack不等式，由此得到平坦性提升结果，还证明了对应线性化方程的Schauder估计。这两类估计均为全新结果，具有独立研究价值。
英文摘要
This paper studies the fully nonlinear Alt-Phillips free boundary problem \begin{equation} F(D^2u) = u^γχ_{\{u>0\}}, \end{equation} for $γ\in (-1/3,1)$ without any convexity assumption on $F$. We establish that flat free boundaries are smooth. This result is new even for the fully nonlinear obstacle problem when $γ= 0$. Our approach is based on a partial hodograph transform, which converts the free boundary into a fixed boundary and produces a fully nonlinear degenerate equation. For this equation we prove a Harnack inequality, yielding an improvement of flatness, and a Schauder estimate for the associated linearized equation. Both estimates appear to be new and are of independent interest.
