On the relaxation of some types of Dirichlet minimum problems for unbounded functionals

In this paper, considered a Borel function g on Rⁿ taking its values in [0,+∞], verifying some weak hypothesis of continuity, such that (domg)^{o}=∅ and domg is convex, we obtain an integral representation result for the lower semicontinuous envelope in the L¹(Ω)-topology of the integral functional G⁰(u₀,Ω,u)=∫_{Ω}g(∇u)dx, where u∈W_{loc}^{1,∞}(Rⁿ), u=u₀ only on suitable parts of the boundary of Ω that lie, for example, on affine spaces orthogonal to aff(domg), for boundary values u₀ satisfying suitable compatibility conditions and Ω is geometrically well situated respect to domg. Then we apply this result to Dirichlet minimum problems.