Generalized Solutions of First-Order PDEs

Hamilton-Jacobi equations and other types of partial differential equations of the first order are dealt with in many branches of mathematics, mechanics and physics. As a rule, functions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. Thus, there arises the need to introduce a notion of a generalized solution and to develop theory and methods for constructing these solutions. This text presents an approach to partial differential equations that can be considered as a non-classical method of characteristics, according to which the generalized solution (the minimax solution) is assumed to be flow invariant with respect to the so-called characteristic inclusions. The research on minimax solutions employs methods of the theory of differential games, dynamical optimization and nonsmooth analysis. At the same time, this research has contributed to the development of these new branches of mathematics. The book is intended as a self-contained exposition of the theory of minimax solutions. It includes existence and uniqueness results, examples of modelling and applications to the theory of control and differential games.