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

2023/09/12 by Liu, Hairong, Tian, Long, Yang, Xiaoping · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.06111

Abstract

In this paper, we focus on the quantitative unique continuation property of solutions to Δ²u=Vu, where V∈ W^1,∞. We show that the maximal vanishing order of the solutions is not large than C(‖V‖^(1)/(4)_L^∞+‖∇ V‖_L^∞+1). 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.

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

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