Lipschitz regularity for $p$-harmonic interface transmission problems

Abstract

We prove optimal Lipschitz regularity for weak solutions of the measure-valued $p$-Poisson equation $-\hat{I}{”}_{p}u=Q{\mathcal{H}}^{n-1}└\hat{I}“$. Here $p\in (1,2)$, $\hat{I}“$ is a compact and connected $C^2$-hypersurface without boundary, and $Q$ is a positive $W^{2,\infty }$-density. This equation can be understood as a nonlinear interface transmission problem. Our main result extends previous studies of the linear case and provides further insights on a delicate limit case of (linear and nonlinear) potential theory.

The manuscript contains an error in the proof Lemma 7 that is not fixable. More precisely, in the Young estimate on p. 13 line 8, ${\epsilon }^{p-1}$ must be replaced by ${\epsilon }^{1/(p-1)}$. Since Lemma 7 plays a crucial role in the proof of the main theorem, its proof must be regarded as incomplete in its current form