Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity q>1 and with natural growth

In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension $n=q$ without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets $\Omega_0 \subset \Omega$, $\Sigma_0 \subset \partial \Omega$ are always empty for $n=q$. Moreover we show that also for $1< q < 2$, but $q$ close enough to $2$, the solutions are global Hölder continuous for $n = 2$.