Developments in the Theory of Nonlinear First-Order Partial Differential Equations

Publisher Summary

There has been a substantial development of the theory of scalar, nonlinear, and first-order partial differential equations in the past two years. This chapter discusses the development in the theory of nonlinear first-order partial differential equations. The chapter describes the initial boundary value problem (IBVP) where uniqueness results are also discussed. Approximation and representation of the solutions are also presented in the chapter. The chapter presents the notions of solutions and uniqueness. The existence theory for viscosity solutions of IBVP and boundary value problem (BVP) is much more a continuation of the existence theory that predates the notion of viscosity solutions than the corresponding uniqueness theory is a continuation of more classical results. The chapter reviews the relationship between the notion of a viscosity solution and control theory in the simplest possible case. It considers a finite horizon control problem without the boundary conditions and formulates two typical theorems. The chapter introduces the Hamilton–Jacobi equations that are classically derived in the closely related areas of the calculus of variations, optimal control theory.