Let \(\Omega\subseteq R^d\) be a bounded domain, and let \(\Sigma \subset \Omega\) denote a hypersurface. Suppose that \(\Omega\setminus \Sigma\) has two connected components, \(\Omega_+\) and \(\Omega_-\), and we have coefficients \(a^{\pm }_{jk}\in L^\infty (\Omega_\pm)\) with \(j,k\in \{1,\ldots ,n\}\).
We say the operator
\[ \mathcal{L}u = \sum_{j,k} \partial_j[(𝟙_{\Omega_{-}} a_{jk}^- + 𝟙_{\Omega_+}a_{jk}^+)\partial_k u] \]is elliptic with positive constants \(\lambda\) and \(M\) if
\[ \lambda |\xi|^2 \leq \sum_{j,k}a_{jk}^\pm (x)\xi_j \xi_k \leq \lambda^{-1} |\xi|^2, \qquad \forall x\in \Omega_{\pm }, \forall \xi\in \mathbb{R}^d \]and
\[ \lVert a_{jk}^\pm \rVert_{C^{0,1}(\Omega)}\le M. \]