AI 中文总结
该研究针对特殊拉格朗日曲率势方程的凸图解，引入混合衰退相容性条件建立边界二阶导数估计，明确曲率发散与边界相位差的关系及最优速率。
AI 中文摘要
我们为特殊拉格朗日曲率势方程的凸图解建立了边界二阶导数估计。由于曲率矩阵同时依赖于$Du$和$D^2u$，仅相位次解无法提供混合导数估计所需的完整线性化分离条件。我们引入仅施加于双重退化水平喷流的混合衰退相容性条件，该条件给出了一致的混合导数界，且在本文考虑的固定光滑零阶障碍类中是尖锐的。对于双法向导数，精确的复Schur补恒等式给出：$u_{\nu\nu}=\beta+\beta\cot\delta$，其中$1\leq\alpha\leq C$，$|\beta|\leq C$，$\beta$和$\beta$是显式Schur补系数，$\beta$是实际边界极限相位差，$C$仅依赖于梯度和混合边界导数的一致界。因此，曲率当且仅当该差消失时发散，最优速率为$\beta^{-1}$。光滑径向解达到该速率，而秩损失模型表明严格下解不一定强制严格凸性。
英文摘要
We establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation. Since the curvature matrix depends on both $Du$ and $D^2u$, a phase subsolution alone does not provide the full linearized separation needed for the mixed derivative estimate. We introduce a mixed recession compatibility condition imposed only on doubly degenerate level jets. It yields a uniform mixed derivative bound and is sharp within the class of fixed smooth zero-order barriers considered here. For the double-normal derivative, an exact complex Schur-complement identity gives \[ u_{νν}=β+α\cotδ, \qquad 1\leqα\leq C, \qquad |β|\leq C, \] where $α$ and $β$ are explicit Schur-complement coefficients, $δ$ is the actual boundary limiting-phase gap, and $C$ depends only on uniform bounds for the gradient and the mixed boundary derivatives. Thus curvature blows up if and only if this gap collapses, with optimal rate $δ^{-1}$. Smooth radial solutions attain the rate, while a rank-loss model shows that a strict lower subsolution need not force strict convexity.
Comments32 pages, 2 figures
