On the Γ-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound

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In Part I we construct the upper bound, in the spirit of Γ-, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E(v):=∫Ω1/(ⁿ∇ⁿ v,...,∇ v,v)dx v:Ω⊂ⁿ→ᵏ such that A·∇ v=0, where the function F≥ 0 and A:ᵏ× ⁿ→ᵐ is a prescribed linear operator (for example, A:≡ 0, A·∇ v:=curlv and A·∇ v=divv) which includes, in particular, the problems considered in [27]. This bound is in general sharper then one obtained in [27].