Integrability for solutions to some anisotropic elliptic equations (Q412768)

The authors consider the boundary value problem NEWLINE\[NEWLINE \begin{cases} \sum_{i=1}^nD_i(a_i(x,Du(x)))=0, & x\in\Omega\cr u(x)=u_\ast(x) & x\in \partial\Omega. \end{cases} NEWLINE\]NEWLINE It is shown that higher integrability of the boundary datum \(u_\ast\) forces solutions \(u\) to have higher integrability as well. Assumptions on \(a_i(x,z)\) are suggested by the Euler equation of the anisotropic functional NEWLINE\[NEWLINE \int_\Omega (|D_1u|^{p_1}+|D_2u|^{p_2}+\ldots+ |D_nu|^{p_n} )dx NEWLINE\]NEWLINE that is NEWLINE\[NEWLINE |a_i(x,z)|\leq C (1+|z_i|)^{p_i-1}. NEWLINE\]