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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, $Q\inC^{0,\alpha}$ is a density on $\Gamma$, and the coefficient matrix $A$ issymmetric, uniformly elliptic and $W^{1,q}$-regular $(q > n)$. We also discussoptimality of these assumptions on the data. The equation can be understood asa special coupling of two $A$-harmonic functions with an interface $\Gamma$. Assuch it plays an important role in several free boundary problems, as we shalldiscuss.

On elliptic equations involving surface measures