Abstract (EN)

This chapter analyzes viscosity solutions of Hamilton–Jacobi equations (HJEs) in Banach spaces. The chapter considers HJEs of the form H(x, u, Du) = 0 in Ω where Ω is an open subset of a Banach space V, V* is the dual of V, H ∈ C(V × R × V*), and Du denotes the Frechet derivative of a function u: Ω → R. A function u is a classical solution of (HJ) in Ω if u is continuously Frechet differentiable on Ω and the equation is satisfied pointwise. The chapter describes that even if V = Rn, the notion of a classical solution is too restrictive to admit the “solutions” of HJEs that are important in the areas in which they arise—in particular, the “value” functions of control theory, the calculus of variations, and differential games that are usually nonclassical solutions of HJEs. The chapter sketches the basic definitions and some existence and uniqueness theorems in Banach spaces. The stability property of the class of viscosity subsolutions in regular spaces is also discussed in the chapter.