On improved regularity of weak solutions of some degenerate, anisotropic elliptic systems (Q678160)

The authors consider integral functionals of the type \(I(u)=\int_\Omega F(Du(x))dx\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(u\) is a vector valued function from \(\Omega\) into \(\mathbb{R}^N\), \(F\) satisfies an anisotropic growth condition of the form \(a \sum_{i=1}^{n}|\xi_i|^{q_i}-b\leq F(\xi) \leq c \sum_{i=1}^{n}|\xi_i|^{q_i}+d\) for every \(\xi\in \mathbb{R}^{nN}\), \(a\), \(b\), \(c\), \(d\) are positive constants, and \(q_i>1\). In the particular case when \(q_i=2\) for \(i=1,...,n-1\), \(q_n=p\) with \(2-2/n<p<2\), and under additional conditions on \(F\), it is proved that any minimizer \(u\) for \(I\) in \(W^{1,p}\) such that \(D_i u\in L^2\) for \(i=1,...,n-1\), satisfies \(D_n u\in L^2_{loc}\). If moreover \(2-2/(n+1)<p<2\), it is also proved that \(D(D_iu)\in L^2_{loc}\) for \(i=1,...,n-1\) and \(D(D_nu)\in L^p_{loc}\). Some other results of the same kind are also given. Proofs are mainly based on the so-called translation method. These results are closely related to those by \textit{P. Marcellini} [Arch. Ration. Mech. Anal. 105, No. 3, 267-284 (1989; Zbl 0667.49032)].