Leszek S. Zaremba

Copyright © 1993 Leszek S. Zaremba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study Isaacs' equation (∗)wt(t,x)+H(t,x,wx(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 Wx(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 Wx(t,x). Using ideas from a very recent paper of Subbotin, 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 Γ. We conclude with providing a close relationship of the results presented on Isaacs' equation with a differential games theory.