R. J E Se(A) for each 1 :::; j :::; n, and in this case we write: IE Se(Aj Rn). In addition, we will say that I is strictly differentiable at x E A, in the direction y if, IJ(xj y) = - IJ(xj -y) for each 1 :::; j :::; n.
Given a function ¢ EX, smooth in IR? , it is a Morse function) we define H(t)¢(x) = supp. : A a regular value of ¢, x E H(t)[¢ ~ An. (27) H(t) acts as a order preserving semigroup of contractions which can be extended to all X. Then, following the strategy of [14J we prove that for any ¢ E X, H(t)¢ is a viscosity solution of (20). Then, adapting the usual proof, we prove a comparison result between sub- and supersolutions of (20). Theorem 4 Let u be upper semicontinuous in IR? \Z, v E C([O, T], X).