Strong Uniqueness for Laplace and Bi-Laplace Operators in the Limit Case

Abstract

In this article we study some limiting cases of strong unique continuation for inequalities of the type

$$ \left| {\Delta u\left( x \right)} \right| \leqslant \frac{A} {{\left| x \right|^2 }}\left| {u\left( x \right)} \right| + \frac{B} {{\left| x \right|}}\left| {\nabla u\left( x \right)} \right| x \in \Omega , $$

(1.1)

or

$$ \left| {\Delta ^2 u\left( x \right)} \right| \leqslant \frac{A} {{\left| x \right|^4 }}\left| {u\left( x \right)} \right| + \frac{B} {{\left| x \right|^3 }}\left| {\nabla u\left( x \right)} \right| + \frac{C} {{\left| x \right|^2 }}\sum\limits_{i,j} {\left| {D_i D_j u\left( x \right)} \right|} x \in \Omega , $$

(1.2)

where Ω is a neighbourhood of the origin in R ⁿ, and A, B, C are positive constants.