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.
