On global discontinuous solutions of Hamilton-Jacobi equations
The aim of this paper is to study global discontinuous solutions of the Cauchy problem for the Hamilton-Jacobi equations: \[ \begin{cases} u_t+H(t,x,u, Du)=0,\;x\in\mathbb{R}^d,\;t>0,\\ u(0,x)=\varphi(x). \end{cases}\tag{1} \] Under some natural conditions on \(H\) the authors prove that the classical semicontinuous solution of (1) is unique, providing that \(\varphi(x)\) is almost everywhere continuous (a.e.-continuous, for short). Moreover they show that, in the space of a.e.-continuous functions, (1) is well posed, and they establish the Lipschitz regularity of the discontinuous solutions after finite time. Note that this regularization effect is new even for conventional continuous viscosity solutions.
scientific article; zbMATH DE number 1744363
scientific article