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In part II we constructed the lower bound, in the spirit of $\Gamma$-$\liminf$ for some general classes of singular perturbation problems, with orwithout the prescribed differential constraint, taking the formE_\e(v):=\int_\Omega \frac{1}{\e}F\Big(\e^n\nabla^n v,...,\e\nablav,v\Big)dx\quad\text{for}\;\; v:\Omega\subset\R^N\to\R^k\;\;\text{suchthat}\;\; A\cdot\nabla v=0, where the function $F\geq 0$ and $A:\R^{k\timesN}\to\R^m$ is a prescribed linear operator (for example, $A:\equiv 0$,$A\cdot\nabla v:=\text{curl}\, v$ and $A\cdot\nabla v=\text{div} v$).Furthermore, we studied the cases where we can easy prove the coinciding ofthis lower bound and the upper bound obtained in [33]. In particular we findthe formula for the $\Gamma$-limit for the general class of anisotropicproblems without a differential constraint (i.e., in the case $A:\equiv 0$).

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