Quantitative uniqueness for bi-Laplace equations with $C^2$ lower-order coefficients

Abstract

We study quantitative unique continuation for bi-Laplace equations \[ Δ^2u+W(x)Δu+V(x)\cdot\nabla u=0 \] with $C^2$ lower-order coefficients. Using the weighted-frequency function, we establish an $L^{\infty }$ three-ball inequality whose exponential factor has coefficient dependence $\Vert W{\Vert }_{C^2}^{1/2}+\Vert V{\Vert }_{C^2}^{1/3}$. Under some normalization, we obtain a vanishing-order bound of size $C(\Vert W{\Vert }_{C^2}^{1/2}+\Vert V{\Vert }_{C^2}^{1/3})$.

arXiv comment29 pages