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

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In part II we constructed the lower bound, in the spirit of Γ- 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→ᵏsuch 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:=curl v and A·∇ v=div v). Furthermore, we studied the cases where we can easy prove the coinciding of this lower bound and the upper bound obtained in [33]. In particular we find the formula for the Γ-limit for the general class of anisotropic problems without a differential constraint (i.e., in the case A:≡ 0).