Energy estimates of harmonic maps between Riemannian manifolds

Let $\mathrm{\Omega }\subset R^n,$ $n\ge 3,$ be a bounded open set, $x=(x_1,x_2,\dots ,x_n)$ a generic point which belongs to $\mathrm{\Omega },$ $u \colon \Omega \to {R}^N ,$ $N>1,$ and $Du=(D_{\alpha }{u}^i)$, $D_\alpha = \partial/\partial x_\alpha, $ $\alpha =1,\dots ,n,$ $i=1,\dots ,N\mathrm{.}$ Main goal is the study of regularity of the minima of nondifferentiable functionals $$\mathcal{F}={\int }_{\mathrm{\Omega }}F(x,u,Du)dx\mathrm{.}$$ 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.