Nonsmooth analysis approach to Isaac's equation

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 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.