有界凸区域中受限半拉普拉斯算子的扭转函数的严格凹性
Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
- University of Science and Technology of China（中国科学技术大学）
- Harbin Normal University（哈尔滨师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明有界凸区域中受限半拉普拉斯算子的扭转函数的二阶导数矩阵处处负定，通过反射调和延拓、连续性方法等完成论证，为相关偏微分方程理论提供关键结论。
AI中文摘要：
设D是n维欧氏空间ℝⁿ（n≥2）中的有界凸区域，u_D为受限半拉普拉斯算子的扭转函数。我们证明在D内每一点，二阶导数矩阵D²u_D均为负定。论证基于狭缝区域中的反射调和延拓，狭缝区域的定量Schauder估计给出边缘余项一阶与二阶导数的参数一致估计；随后通过Schur补计算确定狭缝边缘附近延拓Hessian的惯性。对数Hessian行列式的超调和性与Gleason-Wolff零集定理排除内部退化，从单位球出发的连续性方法证明光滑一致凸区域的结果，穷举法处理任意有界凸区域。
英文摘要：
Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.
