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

2023/09/17 by Hairong Liu, Xiaoping Yang, Liu, Hairong +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2309.09172

openalex publication_date 2023/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation Δ²_Xu=Vu, where Δ_X=Δ_x+|x|^2αΔ_y (0<α≤1), with x∈ℝ^m, y∈ℝⁿ, denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V|≤ (c₀)/(ρ⁴), where ρ=(|x|^2(α+1)+(α+1)²|y|²)^{(1)/(2(α+1))} 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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• Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential

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