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 Δ2y=V0y+V1·∇y+V2Δy+V3·∇Δy in some open, connected set Ω of double-struckRn with V0,0.1emV2∈L∞false(normalΩ;double-struckCfalse) and V1,0.1emV3∈L∞false(normalΩ;0.1emdouble-struckCnfalse). 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 Cfalse|V0false|∞13+false|V1false|∞12+false|V2false|∞23+false|V3false|∞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 Vj and y, we prove that the nontrivial solution y satisfies the decay property e−CR2logR around the point at infinity. In particular, if V1=V3=false(0,…0.1em,0false), this decaying rate can be improved to e−CR4false/3logR.