Relaxation in an L^∞-optimization problem

Let Ω be an open bounded subset of ℝⁿ and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫_Ω f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is 'achieved' by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integral ∫_Ω [1/pH(Du(x))^p - f(x)u(x)] dx. An asymptotic problem as k → ∞ for inf ∫_Ω [k dist(Du(x), K) - f(x)u(x)] dx is also considered, where the infimum is taken all over u ∈ W₀ ^1,1(Ω) and the set K is given by {ξ