On the Γ-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 Γ- limsup, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form
Eε(v):=∫Ω1εF(εn∇nv,…,ε∇v,v)dx∀v:Ω⊂RN→Rks.t.A⋅∇v=0,
where the function F≥0 and A:Rk×N→Rm is a prescribed linear operator (for example, A:≡0, A⋅∇v:=curlv, and A⋅∇v=div v) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17]. v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17].