A frequency function approach to quantitative unique continuation for elliptic equations
Blair Davey
TL;DR
This work develops a frequency-function framework to obtain quantitative unique continuation results for second-order elliptic equations with lower-order terms, focusing on the generalized Schrödinger operator $L=-\mathrm{div}(A \nabla u)+W\cdot \nabla u+V u$ with $A$ Lipschitz, $W\in L^\infty$, $V\in L^p$, $p\ge n$. By constructing Almgren-type frequency functions and establishing their (almost) monotonicity, the authors derive three-ball inequalities that yield explicit lower bounds on vanishing order and Landis-type decay at infinity. The paper proves four main theorems—two local order-of-vanishing results and two Landis-type infinity-decay results—in both the classical ($A=I$, $V\in L^\infty$) and generalized, variable-coefficient settings, with precise dependencies on $K$, $M$, $\eta$, and $p$. Importantly, these results are obtained via a Carleman-free, frequency-function method that extends to variable coefficients through the introduction of $\mu$ and $Z$ and a refined monotonicity analysis. The approach provides a unified, explicit, and potentially sharp framework for quantitative unique continuation in elliptic PDEs with rough coefficients and lower-order terms.
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 Schrödinger 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.