Global summability for solutions to some anisotropic elliptic systems (Q888829)

This paper is devoted to the study of the following anisotropic elliptic system of \(N\) partial differential equations \[ \begin{cases} \sum_{i=1}^n D_i(a_i^{\alpha}(x, Du(x)))=0,\quad & x\in \Omega,\\ u(x)=u_*(x),\quad & x\in \partial\Omega,\end{cases}\tag{1} \] where \(u: \Omega \rightarrow \mathbb R^N, \Omega \) is a bounded open subset of \(\mathbb R^n\) and \(a_i^{\alpha}: \Omega \times \mathbb R^{N\times n} \rightarrow \mathbb R\), \(1\leq \alpha \leq N\), with \( x \mapsto a_i^{\alpha} (x; z)\) measurable and \( z \mapsto a_i^{\alpha} (x; z)\) continuous. By assuming suitable ellipticity and growth conditions on \(a_i^{\alpha}\), the authors show that high degree of integrability of boundary datum \(u_*\) improves the integrability of the solution \(u\) of problem \((1)\). Moreover, with further Lipschitz continuity and monotonicity conditions, they improve the degree of integrability of the solution.