Everywhere Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank one integrands (Q2684471)

In the paper under review, the author studies a class of vectorial problems. Particulary, the regularity of the local minima of the following integral functional is obtained: \[ J(u,\Omega)=\int_\Omega \sum_{\alpha=1}^{n}f_\alpha(x,u^\alpha (x),\nabla u^\alpha (x)) +G(x,u(x),\nabla u (x))dx, \] where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\) and \(u\in W^{1,p}(\Omega,\mathbb{R}^n)\) with \(N\geq 2\), \(n\geq 1\) and \(1<p<N\). Indeed, the author proves that if \(u\) is a local minimizer of the functional \(J(u,\Omega)\) then every component \(u^\alpha\) of the vectorial function \(u\) is a locally Hölder continuous function.