Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential
arXiv (Cornell University)(2023)
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.
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Key words
subelliptic operators,strong unique continuation property,baouendi-grushin
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