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Problems involving Hamilton-Jacobi equationsâwhich we take to be either of the stationary form H (x, u, Du) = 0 or of the evolution form uₓ + H (x, t, u, Du) = 0, where Du is the spatial gradient of uâarise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.
Crandall et al. (Sun,) studied this question.