Let be a bounded open subset of , be the set of real symmetric matrices and . The main result of this paper is the following theorem. Let u be bounded and upper-semicontinuous and v be bounded and lower-semicontinuous. Let and satisfy
Then there are such that
and , , , , where denotes the closure of the set of second order superdifferentials (respectively, sub-differentials) of u (respectively, v) at (respectively, ). Moreover, there is a such that (1) holds with and . (Here orderings are in the sense of quadratic forms).
From this theorem the author obtains comparison result for viscosity solutions of fully nonlinear second order elliptic equations. He also formulates a version of the above theorem appropriate to the discussion of fully nonlinear parabolic equations.