Lipschitz regularity for Poisson equations involving measures supported on $C^{1,\mathrm{Dini}}$ interfaces

We prove optimal Lipschitz regularity of solutions to Poisson’s equation with measure data supported on a $C^{1,\mathrm{Dini}}$ interface and with $C^{0,\mathrm{Dini}}$ density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green’s function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions.