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

2011/12/10 by Poliakovsky, Arkady
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1112.2305

Abstract

In Part I we construct the 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 E_\e(v):=∫_Ω(1)/(\e)F(\eⁿ∇ⁿ v,...,\e∇ v,v)dx\quadfor v:Ω⊂\R^N→\R^k such that A⋅∇ v=0, where the function F≥ 0 and A:\R^{k× N}→\R^m 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 [27]. This bound is in general sharper then one obtained in [27].

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