A Remark on Global $W^{1,p}$ Bounds for Harmonic Functions with Lipschitz Boundary Values

TL;DR

This paper demonstrates that harmonic functions with Lipschitz boundary data have gradients that are pointwise bounded by a universal function belonging to all $L^p$ spaces for finite $p$, providing a new regularity insight.

Contribution

It establishes a universal pointwise bound for the gradient of harmonic functions with Lipschitz boundary values, extending regularity results in harmonic analysis.

Findings

Gradient of harmonic functions is pointwise bounded by a universal $L^p$ function.

The universal bounding function belongs to all $L^p$ spaces for finite $p$.

Provides a new regularity estimate for harmonic functions with Lipschitz boundary conditions.

Abstract

In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.