  We show optimal Lipschitz regularity for very weak solutions of the
(measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \;
\mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset
\mathbb{R}^n$. Here $\Gamma$ is a $C^{1,\alpha}$-regular hypersurface, $Q\in
C^{0,\alpha}$ is a density on $\Gamma$, and the coefficient matrix $A$ is
symmetric, uniformly elliptic and $W^{1,q}$-regular $(q > n)$. We also discuss
optimality of these assumptions on the data. The equation can be understood as
a special coupling of two $A$-harmonic functions with an interface $\Gamma$. As
such it plays an important role in several free boundary problems, as we shall
discuss.
