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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 $u \in Lip(B_{1/2})$

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