接触型的广义哈密顿梯度流I：基本解的正则性
Generalized Hamiltonian gradient flow of contact type I: regularity of the fundamental solutions
AI总结：
本文针对接触型哈密顿-雅可比方程，利用赫尔格洛茨原理的变分框架研究其广义哈密顿梯度流，推导基本解的一阶灵敏度关系与二阶正则性估计，为相关分析方法奠定基础并具多领域应用意义。
AI中文摘要：
本文研究接触型哈密顿-雅可比方程的广义哈密顿梯度流，采用赫尔格洛茨（Herglotz）原理的变分框架及其基本解\nh_L(t,x,y,u)，给出两项主要结果：其一，推导了精确的一阶灵敏度关系，将h_L的导数与极小化轨道的对偶弧联系起来；其二，定量二阶估计表明，h_L是局部半凹的，且在短时间内为半凸的，在某些变量上甚至是一致凸的。这些正则性性质源于对极小化轨道的详细变分分析。该工作为分析接触语境下的奇异性传播和广义梯度流的内禀方法奠定了基础，对弱KAM理论、最优传输及粘性解的正则性具有重要意义。
英文摘要：
This paper studies generalized Hamiltonian gradient flows for contact-type Hamilton--Jacobi equations, adopting the variational framework of Herglotz's principle and its fundamental solution \(h_L(t,x,y,u)\). Two main results are presented. First, precise first-order sensitivity relations are derived, linking derivatives of \(h_L\) to dual arcs of minimizing trajectories. Second, quantitative second-order estimates show that \(h_L\) is locally semiconcave and, over short time, semiconvex, indeed uniformly convex in certain variables. These regularity properties follow from a detailed variational analysis of minimizing trajectories. The work establishes a foundation for intrinsic methods in analyzing singularity propagation and generalized gradient flows in contact context, with implications for weak KAM theory, optimal transport, and regularity of viscosity solutions.
