哈密顿-雅可比方程的割迹 I：通过接触方法的结构与传播

中文摘要 AI 辅助

对于具有线性模的半凹函数，我们通过接触方法引入割迹，并证明其与通过Lax-Oleinik半群定义的变分割迹一致，且与哈密顿量无关。割点的特征是φ的次微分为空，我们将其作为接触定义背后的分析准则。我们引入正则度R_φ，用于衡量φ的C^{1,1}偏差，并证明与割时间函数τ_{φ,H}(x)满足点态下带显式截断常数的定量关系R_φ(x)~1/τ_{φ,H}(x)。由此关系我们推导出校准曲线的分离估计，表明在割迹前不会出现共轭点。该估计为传播结果提供了工具：割点沿演化哈密顿-雅可比方程的广义特征全局传播，亚历山德罗夫点沿校准曲线向前传播，且二阶导数满足矩阵黎卡提方程。我们还证明割迹是勒贝格零集，并给出亚历山德罗夫定理的简化证明。最后，我们基于割时间函数、C^{1,1}支撑及正则度，给出弱KAM解的割迹为闭集的充要条件。

英文摘要

For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax--Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of $ϕ$, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity $R_ϕ$, which measures the $C^{1,1}$ deviation of $ϕ$, and prove the quantitative relation $R_ϕ(x)\sim 1/τ_{ϕ,H}(x)$ with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton--Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the $C^{1,1}$ support, and the degree of regularity.

发表机构

• Università di Roma “Tor Vergata”（罗马托尔维加塔大学）
• Nanjing University（南京大学）
• Nanjing University of Aeronautics and Astronautics（南京航空航天大学）

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