AI 中文总结
研究一类退化完全非线性椭圆方程的粘性解的赫尔德正则性，该方程算子仅在黑塞矩阵大处椭圆，结合修正尖点函数与接触集分解证明，得出自然假设下的内部赫尔德连续性结论。
AI 中文摘要
本文为一类退化的完全非线性椭圆方程\[ F(D^2u,Du)=f(x)~~\text{在}~~B_1中\]的粘性解建立了赫尔德正则性，其中算子仅在黑塞矩阵足够大的区域是椭圆的。此类方程自然出现于自由边界问题和部分椭圆性模型中。证明结合了修正的尖点函数与接触集分解，在自然结构假设下得出内部赫尔德连续性。
英文摘要
In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form \[ F(D^2u,Du)=f(x)~~\text{in}~~B_1, \] where the operator is elliptic only in regions where the Hessian is sufficiently large. Such equations arise naturally in free boundary problems and models with partial ellipticity. The proof combines a modified cusp function with a decomposition of the contact set in a point-to-measure argument. As a consequence, interior Hölder continuity follows under natural structural assumptions.
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