A frequency function approach to quantitative unique continuation for elliptic equations

TL;DR

This paper develops a new frequency function-based method to analyze quantitative unique continuation for elliptic equations, extending previous Carleman estimate approaches and addressing Landis' conjecture on decay rates.

Contribution

It introduces novel proof techniques using frequency functions to establish quantitative and global unique continuation properties for elliptic equations with lower-order terms.

Findings

Established quantitative strong unique continuation estimates.

Proved results related to Landis' conjecture on decay at infinity.

Provided new methods avoiding Carleman estimates.

Abstract

We investigate the quantitative unique continuation properties of solutions to second-order elliptic equations with lower-order terms. In particular, we establish quantitative forms of the strong unique continuation property for solutions to generalized Schr\"odinger equations of the form $- \text{div}(A \nabla u) + W \cdot \nabla u + V u = 0$, where we assume that $A$ is bounded, elliptic, symmetric, and Lipschitz continuous, while $W$ belongs to $L^\infty$ and $V$ belongs to $L^p$ for some $p \ge n$. We also study the global unique continuation properties of solutions to these equations, establishing results that are related to Landis' conjecture concerning the optimal rate of decay at infinity. Versions of the theorems in this article have been previously proved using Carleman estimates, but here we present novel proof techniques that rely on frequency functions.