Quantitative uniqueness of some higher order elliptic equations

TL;DR

This paper investigates the quantitative unique continuation properties of higher order elliptic equations, providing decay bounds at infinity and constructing sharp examples in two dimensions to demonstrate the limits of these bounds.

Contribution

It establishes lower bounds for decay at infinity for solutions of higher order elliptic equations, including sharp bounds in two dimensions with explicit examples.

Findings

Derived lower bounds of decay at infinity for solutions of $(- abla)^m$ operators.

Constructed Meshkov type examples showing sharpness of bounds in 2D.

Extended understanding of unique continuation for higher order elliptic equations.

Abstract

We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-\Delta)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under some general assumptions. Furthermore, in dimension 2, we can obtain essentially sharp lower bounds for some forth order elliptic operators, the sharpness is shown by constructing a Meshkov type example.