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

TL;DR

This paper establishes optimal Lipschitz regularity for solutions to Poisson's equation with measure data on $C^{1, ext{Dini}}$ interfaces, using gradient estimates and Green's function analysis.

Contribution

It introduces new pointwise gradient estimates and demonstrates piecewise differentiability of solutions up to $C^{1, ext{Dini}}$ interfaces, with sharp counterexamples.

Findings

Optimal Lipschitz regularity proven for solutions.

Solutions are piecewise differentiable up to the interface.

Counterexamples show minimality of assumptions.

Abstract

We prove optimal Lipschitz regularity of solutions to Poisson's equation with measure data supported on a $C^{1,\operatorname{Dini}}$ interface and with $C^{0,\operatorname{Dini}}$ 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.