In this paper, we prove a three-ball inequality for y satisfying an equation of the form

$${\mathrm{\Delta }}^{2}y=V_{0}y+V_1·\nabla y+V_2\mathrm{\Delta }y+V_3·\nabla \mathrm{\Delta }y$$

in some open, connected set Ω of ${\mathbb{R}}^n$ with $V_0,V_2\in L^{\infty }(\mathrm{\Omega };\mathbb{C})$ and $V_1,V_3\in L^{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$. 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 $C|V_0{|}_{\infty }^{\frac{1}{3}}+|V_1{|}_{\infty }^{\frac{1}{2}}+|V_2{|}_{\infty }^{\frac{2}{3}}+|V_3{|}_{\infty }^2$, where | · |_∞ means the L^∞ norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on V_j 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 $V_1=V_3=(0,\dots ,0)$, this decaying rate can be improved to $e^{-C{R}^{4/3}\log R}$.

中文翻译：

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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

$${\mathrm{\Delta }}^2是={伏}_0是+{伏}_1·\nabla 是+{伏}_2\mathrm{\Delta }是+{伏}_3·\nabla \mathrm{\Delta }是$$

在一些开放的，连接的集合Ω中 ${\mathbb{电阻}}^n$ 和 ${伏}_0,{伏}_2\in {升}^{\infty }(\mathrm{\Omega };\mathbb{C})$ 和 ${伏}_1,{伏}_3\in {升}^{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$. 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$C|{伏}_0{|}_{\infty }^{\frac{1}{3}}+|{伏}_1{|}_{\infty }^{\frac{1}{2}}+|{伏}_2{|}_{\infty }^{\frac{2}{3}}+|{伏}_3{|}_{\infty }^2$，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 _j和y 的一些先验假设下，我们证明了非平凡解y满足衰减性质${\mathrm{电子}}^{-C{\mathrm{电阻}}^2\mathrm{日志}\mathrm{电阻}}$在无穷远点附近。特别地，如果${伏}_1={伏}_3=(0,\dots ,0)$，这个衰减率可以提高到 ${\mathrm{电子}}^{-C{\mathrm{电阻}}^{4/3}\mathrm{日志}\mathrm{电阻}}$.