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

Abstract

We study quantitative unique continuation for bi-Laplace equations \[ \Delta^2u+W(x)\Delta u+V(x)\cdot\nabla u=0 \] 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 $\|W\|_{C^2}^{1/2}+\|V\|_{C^2}^{1/3}$. Under some normalization, we obtain a vanishing-order bound of size $C(\|W\|_{C^2}^{1/2}+\|V\|_{C^2}^{1/3})$.