Energy estimates of harmonic maps between Riemannian manifolds

Let $Ω\subset {R}^n,$ $n \geq 3,$ be a bounded open set, $x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $Ω,$ $u \colon Ω\to {R}^N ,$ $N>1,$ and $Du=(D_αu^i)$, $D_α= \partial/\partial x_α,$ $α=1,\ldots,n,\,$ $i=1,\ldots,N .\,$ Main goal is the study of regularity of the minima of nondifferentiable functionals ${\cal F} \,=\, \int_ΩF(x,u,Du) dx.$ 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.

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