On viscosity solutions of irregular Hamilton-Jacobi equations

摘要

It is proved thai the initial-value problem for u_t + H(x, ▽_u) = 0 admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x,p) is discontinuous in x. Particular attention is devoted to the linear transport equation u_t + a(x)·▽_u = 0, where a may be discontinuous.