发表机构
Universidade Federal da Paraíba; Oklahoma State University（帕拉伊巴联邦大学; 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
本文通过谱分辨接触几何机制，解决了平面及三维全非线性椭圆上解的尖锐Hessian可积性猜想，给出了最优指数并识别了剩余差距的谱配置。
AI 中文摘要
我们解决了Armstrong、Silvestre和Smart提出的平面尖锐Hessian可积性猜想（\textit{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184），该猜想针对全非线性一致椭圆方程的粘性上解。若$\kappa=\Lambda/\lambda$，则平面中$W^{2,\epsilon}$正则性理论的最优指数恰为$$ \epsilon_2(\kappa)=\frac{2}{\kappa+1}. $$ 同一机制在三维情形下于整个范围$1\le \kappa\le 4$内达到已知的上界障碍，从而给出$$ \epsilon_3(\kappa)=\frac{3}{2\kappa+1} \qquad(1\le\kappa\le4). $$ 超过此阈值后，它产生一个封闭的代数下界，并识别出导致剩余差距的精确谱配置。证明引入了一种接触几何的谱分辨连续开口机制。在整个接触演化过程中，顶点雅可比行列式保留Hessian的负指数及其平均负曲率——这些数据被经典逐点约化所抹去——并且这些变量仅在所有开口上积分后才被优化。这将对问题的分析归结为有限族显式一维核，其临界指数恢复了尖锐平面阈值，识别了上述精确三维区域，并揭示了从接触集的谱几何提取二阶正则性的更广泛原理。
英文摘要
We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $κ=Λ/λ$, then the optimal exponent in the $W^{2,ε}$ regularity theory in the plane is exactly $$ ε_2(κ)=\frac{2}{κ+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\le κ\le 4$ and therefore gives $$ ε_3(κ)=\frac{3}{2κ+1} \qquad(1\leκ\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.
Comments26 pages
