Аппроксимация многозначного решения уравнения Гамильтона-Якоби

The paper deals with the construction of a multivalued solution of the Cauchy problem for the Hamilton-Jacobi equation with discontinuous Hamiltonian with respect to the phase variable. The constructed multivalued solution is approximated by a sequence of continuous solutions of auxiliary Cauchy problems of the Hamilton-Jacobi equation with Hamiltonian which is Lipschitz with respect to the phase variable. The results of the study are illustrated by an example.