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 $C<sup>2$ lower-order coefficients. Using the weighted-frequency function, we establish an $L<sup>\infty$ three-ball inequality whose exponential factor has coefficient dependence $$|W|<em>{C^{2}^{1/2}+|V|}</em>{C<sup>2}<sup>{1/3}$$. Under some normalization, we obtain a vanishing-order bound of size $$C(|W|<em>{C^{2}^{1/2}+|V|}</em>{C<sup>2}<sup>{1/3})$$.