发表机构
King Abdullah University of Science and Technology (KAUST); Oklahoma State University（阿卜杜拉国王科技大学; 俄克拉荷马州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立将定性光滑性转化为完全非线性椭圆方程一致定量正则性估计的一般机制，证明紧类中定性定量正则性阈值相同，并给出新的部分正则性判据。
AI 中文摘要
我们建立了一个一般机制，将定性光滑性转化为完全非线性椭圆方程的一致、定量正则性估计。核心结论是，对于由方程的自然缩放保持的紧类，定性正则性与定量正则性具有相同的临界阈值。在二阶情形，仅在相应的中心凸包上处处二次可微就足以推出对某个 $\alpha>0$ 的一致 $C^{2,\alpha}$ 理论。单侧切向版本给出了单个算子新的部分正则性判据：解的每个二次可微点都是正则点，而奇异集具有普遍的正余维数。证明结合了完全重正规化类的紧性与尺度自适应的平坦性改进，将每个剖面上分别可用的信息转化为整个类上的一致估计。该论证不需要对所假设的光滑性进行定量控制，并为光滑性可见但估计仍难以获得的问题开辟了道路。
英文摘要
We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is that, for compact classes preserved by the natural rescalings of the equation, qualitative and quantitative regularity have the same critical threshold. At second order, mere twice differentiability throughout the corresponding centered hull already forces a uniform $C^{2,α}$-theory for some $α>0$. A one-sided tangent version gives a new partial regularity criterion for a single operator: every point at which a solution is twice differentiable is regular, whereas the singular set has universal positive codimension. The proof combines compactness of the full renormalized class with a scale-adaptive improvement of flatness, converting information available separately at each profile into estimates uniform across the entire class. The argument requires no quantitative control of the assumed smoothness and opens a path in problems where smoothness is visible, but estimates remain out of reach.
