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We model shortest paths in cracked domains using a Hamilton-Jacobi equation with a state constraint on the crack, formulated as a variational inequality where the crack acts as a two-sided barrier. Trajectory-based arguments establish a comparison principle and uniqueness for the value function, bypassing limitations of the classical doubling technique. The variational inequality yields a free boundary on the crack, defining sliding and illuminated regimes with a geometric characterization including grazing contact and diffraction at crack tips.
Optimal trajectories in cracked domains exhibit distinct sliding and illuminated regimes, with crack tips acting as diffraction centers.
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.