Quantitative unique continuation of solutions to the bi‐Laplace equations

Abstract

In this paper, we prove a three‐ball inequality for y satisfying an equation of the form in some open, connected set Ω of with and . 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 , 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 around the point at infinity. In particular, if , this decaying rate can be improved to .