In this paper we construct upper bounds for families of
functionals of the form
$$
E_\varepsilon(\phi):=\int_\Omega\Big(\varepsilon |\nabla\phi|^2+\frac{1}{\varepsilon }W(\phi)\Big){\rm d}x+\frac{1}{\varepsilon }\int_{{\mathbb{R}}^N}|\nabla \bar H_{F(\phi)}|^2{\rm d}x
$$
where Δ$\bar H_u$ = div {$\chi_\Omega$u}. Particular cases of such functionals arise in
Micromagnetics. We also use our technique to construct upper bounds
for functionals that appear in a variational formulation of
the method of vanishing viscosity for conservation laws.