Boundary singularities and characteristics of Hamilton–Jacobi equation

Boundary-value problems for first order PDEs are locally considered, when the classical sufficient condition for the solution existence does not hold, but a solution still exists, possibly defined on one or both sides of the boundary surface. We note three situations when such a surface (locally) arises: (1) the part of the boundary surface with the given boundary value; (2) the part of the boundary surface with no value initially specified on it, while such a value arises during the constructions; (3) a singular surface arising during the constructions in the internal part of the domain. In the latter two cases, which are typical for the problems of optimal control and differential games, the solution value on the surface is specified due to singular characteristics. Although the character of the singularity of the solution is well known, we show in this paper that it is completely determined by the signs of two multiple Poisson brackets naturally arising in the equations of singular characteristics. We recall the notion of regular and singular characteristics and formulate a new sufficiency condition for the existence and uniqueness in the irregular case. This condition is in invariant form based on Poisson (Jacobi) bracket, which is convenient in applications. We give several examples with irregular solutions and boundary characteristics