Nonsmooth analysis approach to Isaac's equation (Q1210448)

Summary: We study Isaacs' equation \((*)\) \(w_ t(t,x)+ H(t,x,w_ x(t,x))= 0\) (\(H\) is a highly nonlinear function) whose ``natural'' solution is a value \(W(t,x)\) of a suitable differential game. It has been felt that even though \(W_ x(t,x)\) may be a discontinuous function or it may not exist everywhere, \(W(t,x)\) is a solution of \((*)\) in some generalized sense. Several attempts have been made to overcome this difficulty, including viscosity solution approaches, where the continuity of a prospective solution or even slightly less than that is required rather than the existence of the gradient \(W_ x(t,x)\). Using ideas from a very recent paper of \textit{A. I. Subbotin} [Nonlinear Anal., Theory Methods Appl. 16, No. 7/8, 683-699 (1991; Zbl 0739.35011)], we offer here an approach which, requiring literally no regularity assumptions from prospective solutions of \((*)\), provides existence results. To prove the uniqueness of solutions to \((*)\), we make some lower- and upper-semicontinuity assumptions on a terminal set \(\Gamma\). We conclude with providing a close relationship of the results presented on Isaacs' equation with a differential games theory.