On the F-Limit of singular perturbation problems with optimal profiles which are not one-dimensional part I: The upper bound

In Part I we construct an upper bound, in the spirit of F- lim, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form (eqution presented) where the function F ≥ 0 and A : ℝ^k×N → ℝ^m is a prescribed linear operator (for example, A :Ξ 0, A.∇v := curl v, and A.∇ = div v) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17].