  Let $\Omega \subset {R}^n,$ $n \geq 3,$ be a bounded open set,
$x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $\Omega,$ $u \colon
\Omega \to {R}^N ,$ $N>1,$ and $ Du=(D_\alpha u^i)$, $D_\alpha =
\partial/\partial x_\alpha, $ $\alpha =1,\ldots,n,\,$ $i=1,\ldots,N .\,$
  Main goal is the study of regularity of the minima of nondifferentiable
functionals $$ {\cal F} \,=\, \int_\Omega F(x,u,Du) dx. $$ having the integrand
function different shapes of smoothness. The method is based on the use some
majorizations for the functional, rather than the well known Euler equation
associated to it.
