Harnack Principle for Weakly Coupled Elliptic Systems

We prove the Harnack inequality for the weakly coupled elliptic systemS, where Su=\big (\begin{align} L_1 & & \\ & \cdots & \\ & & L_N\end{align}\big )u+Qu and U=\big (\begin{align} u_1\\ \cdot \\ \cdot \\ \cdot \\ u_N\end{align}\big ). {Lk,k=1, …, N} are second order elliptic operators with Hölder continuous coefficients andQis a matrix-valued function with singular entries. In the case thatQis irreducible, a full Harnack principle is proved.