Potential estimates for the p-Laplace system with data in divergence form

TL;DR

This paper derives pointwise bounds and regularity estimates for solutions to the p-Laplace system with divergence form data, using potential theory and applicable to various function norms.

Contribution

It introduces a unified approach to regularity estimates for the p-Laplace system with divergence form data, extending classical results to broader norm classes.

Findings

Established pointwise bounds involving Havin-Maz'ya-Wulff potentials.

Provided local oscillation bounds for solutions.

Unified regularity framework applicable to multiple norm types.

Abstract

A pointwise bound for local weak solutions to the p-Laplace system is established in terms of data on the right-hand side in divergence form. The relevant bound involves a Havin-Maz'ya- Wulff potential of the datum, and is a counterpart for data in divergence form of a classical result of [KiMa], that has recently been extended to systems in [KuMi2]. A local bound for oscillations is also provided. These results allow for a unified approach to regularity estimates for broad classes of norms, including Banach function norms (e.g. Lebesgue, Lorentz and Orlicz norms), and norms depending on the oscillation of functions (e.g. Holder, BMO and, more generally, Campanato type norms). In particular, new regularity properties are exhibited, and well-known results are easily recovered.