发表机构
Ho Chi Minh City University of Technology (HCMUT); Vietnam National University Ho Chi Minh City（胡志明市理工大学; 越南胡志明市国家大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将含裂纹区域的最短路径问题建模为带状态约束的Hamilton-Jacobi方程，通过轨迹论证证明比较原理与唯一性，并完整刻画滑动、照明及尖端衍射三种最优状态。
AI 中文摘要
我们将含裂纹区域中的最短路径问题表述为一个在裂纹上带有状态约束的Hamilton-Jacobi方程，在裂纹的约束起作用的部位，该方程呈现为变分不等式的形式：裂纹是一个两侧的障碍，可以被接触和沿其移动但不能穿越，因此它作为一个低维状态约束，由其两个唇边分别承载，允许切向传播但禁止跨越。我们通过基于轨迹的论证建立了比较原理和值函数的唯一性，这些论证绕过了Crandall、Ishii和Lions经典加倍技巧在裂纹处的失效。变分不等式在裂纹上诱导出一个自由边界，将滑动状态（最优轨迹沿裂纹运行有限段）与照明状态（裂纹被直接射线横向到达）分开；我们给出了两种状态的完整几何刻画，包括掠射（切向）接触、在正则裂纹点处无反射以及裂纹尖端的衍射定律，所有这些都仅由最优性推导得出。在一个精确可解配置上的正问题模拟重现了这三种状态和尖端衍射定律，精度达到闭式解的精度。
英文摘要
We formulate the shortest-path problem in a domain containing a crack as a Hamilton--Jacobi equation carrying a state constraint on the crack, which on the binding part of the crack takes the form of a variational inequality: the crack is a two-sided barrier that may be touched and followed but not traversed, so that it acts as a lower-dimensional state constraint, carried separately by each of its two lips, which admits tangential propagation but forbids crossing. We establish a comparison principle and uniqueness of the value function through trajectory-based arguments that circumvent the failure of the classical doubling technique of Crandall, Ishii, and Lions at the crack. The variational inequality induces a free boundary on the crack separating a sliding regime, where optimal trajectories run along the crack over a finite segment, from an illuminated regime, where the crack is reached transversally by direct rays; we give a complete geometric characterization of both regimes, including grazing (tangential) contact, the absence of reflection at regular crack points, and a diffraction law at crack tips, all derived from optimality alone. A simulation of the forward problem on an exactly solvable configuration reproduces the three regimes and the tip-diffraction law to the accuracy of the closed-form solution.
Comments43 pages, 4 figures
