  This article is devoted to the study of lower semicontinuous solutions of
Hamilton-Jacobi equations with convex Hamiltonians in a gradient variable. Such
Hamiltonians appear in the optimal control theory. We present a necessary and
sufficient condition for a reduction of a Hamiltonian satisfying optimality
conditions to the case when the Hamiltonian is positively homogeneous and also
satisfies optimality conditions. It allows us to reduce some uniqueness
problems of lower semicontinuous solutions to Barron-Jensen and Frankowska
theorems. For Hamiltonians, which cannot be reduced in that way, we prove the
new existence and uniqueness theorems.
