Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Results on gradients of harmonic functions on Lipschitz surfaces

Benjamin Foster

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.

Results on gradients of harmonic functions on Lipschitz surfaces

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.
Paper Structure (8 sections, 17 theorems, 94 equations)

This paper contains 8 sections, 17 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Conventions and Preliminaries
  3. Empty Superlevel Sets for Nonvanishing Gradients
  4. Holomorphic Polynomial Frequency Control
  5. More General Holomorphic Functions
  6. Propagation of Smallness for Gradients of Harmonic Functions
  7. Proofs of Theorems \ref{['main eff']} and \ref{['main prop']} on Surfaces
  8. More General Elliptic Equations with Lipschitz Coefficients

Key Result

Theorem 1

Suppose $u$ is a solution to $\Delta_g u=0$ on the Euclidean disk of radius 2. Assume its frequency satisfies $N_u(0,1)\le \Lambda$. Then we have the following volume estimate for the superlevel sets of the frequency function where $c<1$ is a suitably chosen constant and where $C>0$ is large enough, both depending on the Lipschitz and ellipticity constants of $g$.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 19 more