Global summability for solutions to some anisotropic elliptic systems

Abstract

We consider the system of $N$ partial differential equations $$ \sum\limits_{i=1}^{n} D_i (a_i^\alpha (x,Du(x))) = 0, \quad x \in \Omega, \quad \alpha \in \{1,...,N\}, $$ and the boundary condition $$ u(x) = u_*(x), \quad x \in \partial\Omega. $$ We show that higher integrability of the boundary datum $u_*$ forces solutions $u$ to have higher integrability as well, provided we assume suitable ellipticity and growth conditions on $a_i^\alpha$.