Quantitative unique continuation property for solutions to a bi-Laplacian equation with a potential

TL;DR

This paper establishes a quantitative measure of how solutions to a bi-Laplacian equation with a potential can vanish, providing explicit bounds related to the potential's norms.

Contribution

It introduces a novel approach by transforming the equation into a system of second-order equations and defining a weighted frequency function to derive explicit vanishing order bounds.

Findings

Maximal vanishing order depends explicitly on the potential's norms.

Decomposition into second-order systems facilitates analysis.

Derived doubling inequalities with explicit potential dependence.

Abstract

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} \Delta^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.