APPLICATIONS OF THE HOPF-LAX FORMULA FOR U(T)+H(U,DU)=0

The gamma (sub)level set of the solution to w(t) + H(gamma, D(x)w) = 0 is the same as the gamma (sub)level set of the solution to u(t) + H(u , D(x)u) = 0, and the solution u may be built from w. This result is a pplied to determining upper and lower bounds for a solution of u(t) H-1(u, Du) + H-2(u, Du) = 0, with H-1 convex and H-2 concave, as well as u(t) + H(u, Du) = 0, but with initial data u(0, x) = g(1)(x) boolea n OR g(2)(x) or g(1)(x) boolean AND g(2)(x), with g(1) quasi-convex an d g(2) quasi-concave. A differential game in L-infinity is constructed giving a new proof of the Hopf formula.