A density approach to Hamilton-Jacobi equations with \(t\)-measurable Hamiltonians (Q2487814)

The authors are studying the so-called ``Ishii's viscosity solutions'' in [\textit{H. Ishii}, Bull. Fac. Sci. Eng., Chuo Univ., Ser. I 28, 33--77 (1985; Zbl 0937.35505)] of problems of the form \[ {{\partial u}\over{\partial t}}+H(t,x,u,\nabla u)=0 \text{ in } (0,T)\times R^N, \quad u(0,x)=\varphi (x), \quad x\in R^N \] in the case the Hamiltonian \(H\) is measurable with respect to the first variable and continuous with respect to the others. The main idea of the paper is to characterize certain continuous approximations, \(H_n\), of \(H\) such that for each sub-(respectively, super-) solution \(u(.,.)\) of the problem, there exists a corresponding object, \(u_n(.,.)\), of the same problem, defined by the (continuous) Hamiltonian \(H_n\), that converges uniformly to \(u(.,.)\). The authors prove that in this context, some basic properties (comparison, uniqueness, existence, regularity) of viscosity solutions for continuous Hamiltonians remain valid for \(t-\)measurable Hamiltonians.