How to prove?
Let $a_{ij} \in C^{0, \alpha}(B_1 \cap \mathbb{R}^{n}_{+}), b_{ij} \in C^{0,\alpha}(B_1 \cap \mathbb{R}^{n}_{-})$ elliptic matrices and $ A_{ij}(x) = a_{ij}(x)\chi_{\{ x_n \ge 0 \}} + b_{ij}(x) \chi_{\{x_n \le 0\}}.$ Show that if $ \mathbb{div}(A_{ij }(x)\nabla u) = 0 \quad \mbox{in} \quad B_1 $ then $u \in \mathrm{Lip}(B_{1/2})$.
Thank you.