Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential

TL;DR

This work addresses quantitative unique continuation for the fourth-order Baouendi-Grushin equation $Δ^2_{X} u = V u$ with the degenerate operator $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$, where $0<β≤1$ and the potential $V$ is bounded with $|ZV|$ controlled by the angle function $ψ$. It adapts Almgren's frequency function to a coupled second-order system by writing $Δ_{X}u = w$ and $Δ_{X}w = V u$, and introduces a weighted height $H(r)$ and energy $I(r)$ to define a generalized frequency $N(r) = I(r)/H(r)$. The authors prove an almost monotonicity of a frequency-like quantity, derive a Hadamard-type three-ball inequality for the height, and obtain explicit bounds on the maximal vanishing order of nontrivial solutions in terms of the potential bounds $K_1$, $K_2$ and the Grushin geometry parameters $(m,n,β)$. These results provide quantitative control over vanishing near the degeneracy set and extend unique continuation theory to fourth-order subelliptic operators in sub-Riemannian settings, with potential implications for control theory and related PDEs. Key formulas include $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$, $ρ(x,y) = (|x|^{2(β+1)} + (β+1)^2 |y|^2)^{1/[2(β+1)]}$, $ψ = |x|^{2β}/ρ^{2β}$, and the decay/vanishing bounds expressed via $K_1$, $K_2$ and the constants arising from the frequency analysis.

Abstract

We investigate the quantitative unique continuation property for solutions to $$Δ^2_{X} u = V u,$$ where $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$ ($0 < β\leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|Z V| \leq K ψ$ for some constant $K>0$, where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (β+1)\sum_{j=1}^n y_j \partial_{y_j}$, $ψ$ is the angle function given by $ψ= \frac{|x|^{2β}}{ρ^{2β}}$, and $$ρ(x,y) = \left(|x|^{2(β+1)} + (β+1)^2 |y|^2\right)^{\frac{1}{2(β+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof