Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential

TL;DR

This paper establishes the strong unique continuation property for a class of fourth order subelliptic operators with singular potentials, using Almgren's frequency function and Hardy-Rellich inequalities.

Contribution

It introduces a novel approach with an Almgren-type frequency function to prove unique continuation for degenerate elliptic equations with strongly singular potentials.

Findings

Proves strong unique continuation property for fourth order Baouendi-Grushin operators.

Develops a monotonicity formula for an Almgren-type frequency function.

Establishes doubling estimates using Hardy-Rellich inequalities.

Abstract

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation \begin{equation*} \Delta^2_{X}u=Vu, \end{equation*} where $\Delta_{X}=\Delta_{x}+|x|^{2\alpha}\Delta_{y}$ ($0<\alpha\leq1$), with $x\in\mathbb{R}^{m}, y\in\mathbb{R}^{n}$, denotes the Baouendi-Grushin type subelliptic operators, and the potential $V$ satisfies the strongly singular growth assumption $|V|\leq \frac{c_0}{\rho^4}$, where \begin{equation*} \rho=\left(|x|^{2(\alpha+1)}+(\alpha+1)^2|y|^2\right)^{\frac{1}{2(\alpha+1)}} \end{equation*} is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.