  In Part I we construct the upper bound, in the spirit of $\Gamma$- $\limsup$,
achieved by multidimensional profiles, for some general classes of singular
perturbation problems, with or without the prescribed differential constraint,
taking the form $$E_\e(v):=\int_\Omega \frac{1}{\e}F\Big(\e^n\nabla^n
v,...,\e\nabla v,v\Big)dx\quad\text{for} v:\Omega\subset\R^N\to\R^k \text{such
that} A\cdot\nabla v=0,$$ where the function $F\geq 0$ and $A:\R^{k\times
N}\to\R^m$ is a prescribed linear operator (for example, $A:\equiv 0$,
$A\cdot\nabla v:=\text{curl}v$ and $A\cdot\nabla v=\text{div}\,v$) which
includes, in particular, the problems considered in [27]. This bound is in
general sharper then one obtained in [27].
