Remarks on the De Giorgi–Moser Techniques for Vectorial Problems

In this paper we study the gradient’s boundedness of the minima of the functional \(\begin{aligned} \int _\Omega (1 +|\nabla u|^2)^{\frac{p}{2}}\,dx \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}{(1+|\mathrm{\nabla }u|}^2{)}^{\frac{p}{2}}dx\end{array}$$ where \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , \(2< p< n\) $2<p<n$ and \(u: \Omega \rightarrow {\mathbb {R}}^m\) $u:\mathrm{\Omega }\to {\mathbb{R}}^m$ . Regularity results are well-known in the literature; however, we hereby employ an alternative technique to prove the boundedness result. The innovative and original aspect of this paper is to employ some new truncation arguments to find new Caccioppoli type inequalities. These inequalities allow for the use of the Moser’s technique in the vectorial case, in order to get the boundedness result.