Results on gradients of harmonic functions on Lipschitz surfaces

TL;DR

This paper investigates the behavior of gradients of harmonic functions on Lipschitz surfaces, providing sharper bounds, a propagation of smallness result, and extending estimates to more general elliptic PDEs.

Contribution

It improves bounds on superlevel sets of the frequency function, develops a propagation of smallness for gradients, and extends estimates to divergence-form elliptic PDEs with bounded drift.

Findings

Sharp quadratic bound on superlevel sets of the frequency function.

Propagation of smallness for gradients of harmonic functions.

Extension of superlevel set estimates to general elliptic PDEs with bounded drift.

Abstract

We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic bound in this setting using complex analytic tools. We also develop a propagation of smallness for gradients of harmonic functions, settling an open question in this setting. Finally, we extend the estimate on superlevel sets of the frequency to more general divergence-form elliptic PDEs with bounded drift terms at the cost of a subpolynomial factor.