Relaxation in an L-infinity-optimization problem

Let Omega be an open bounded subset of Rn and f a continuous function on Omega satisfying f(x) > 0 for all x is an element of Omega. We consider the maximization problem for the integral fOmega f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on partial derivativeOmega and to the gradient constraint of the form H(Du(x)) less than or equal to 1, and prove that the supremum is 'achieved' by the viscosity solution of H(Du(x)) = 1 in Omega and u = 0 on partial derivativeOmega, where H denotes the convex envelope of H. This result is applied to an asymptotic problem, as p --> infinity, for quasi-minimizers of the integral integral(Omega)[1/p H(Du(x))(p) - f(x)u(x)] dx. An asymptotic problem as k --> infinity for inf integral(Omega)[k dist(Du(x),K) - f(x)u(x)] dx is also considered, where the infimum is taken all over u is an element of W-0(1,1)(Omega) and the set K is given by {xi H (xi) less than or equal to 1}