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Quantitative unique continuation of solutions to the bi-Laplace equations
Mathematical Methods in the Applied Sciences ( IF 2.1 ) Pub Date : 2021-04-16 , DOI: 10.1002/mma.7424
Xiaoyu Fu 1 , Zhonghua Liao 1
Affiliation  

In this paper, we prove a three-ball inequality for y satisfying an equation of the form
Δ 2 y = V 0 y + V 1 · y + V 2 Δ y + V 3 · Δ y
in some open, connected set Ω of R n with V 0 , V 2 L ( Ω ; C ) and V 1 , V 3 L ( Ω ; C n ) . The derivation of such estimate relies on a delicate Carleman estimate for the bi-Laplace equation and some Caccioppoli inequalities to estimate the lower-terms. Based on three-ball inequality, we then derive the vanishing order of y is less than C | V 0 | 1 3 + | V 1 | 1 2 + | V 2 | 2 3 + | V 3 | 2 , where | · | means the L norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on Vj and y, we prove that the nontrivial solution y satisfies the decay property e C R 2 log R around the point at infinity. In particular, if V 1 = V 3 = ( 0 , , 0 ) , this decaying rate can be improved to e C R 4 / 3 log R .


中文翻译:

双拉普拉斯方程解的定量唯一延拓

在本文中,我们证明了满足以下形式的方程的y的三球不等式
Δ 2 = 0 + 1 · + 2 Δ + 3 · Δ
在一些开放的,连接的集合Ω中 电阻 n 0 , 2 ( Ω ; C ) 1 , 3 ( Ω ; C n ) . 这种估计的推导依赖于对双拉普拉斯方程的精细卡尔曼估计和一些 Caccioppoli 不等式来估计较低项。基于三球不等式,我们推导出y的消失阶小于 C | 0 | 1 3 + | 1 | 1 2 + | 2 | 2 3 + | 3 | 2 ,哪里| · | 表示L 范数,它是y的强唯一连续属性的量化版本。此外,在V jy 的一些先验假设下,我们证明了非平凡解y满足衰减性质 电子 - C 电阻 2 日志 电阻 在无穷远点附近。特别地,如果 1 = 3 = ( 0 , , 0 ) ,这个衰减率可以提高到 电子 - C 电阻 4 / 3 日志 电阻 .
更新日期:2021-04-16
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