Carleman inequalities and unique continuation for higher-order elliptic differential operators

Abstract

In this thesis, we study the weak unique continuation property for higher order elliptic differential operators with real coefficients via Carleman inequalities. We get several Carleman inequalities with sharp gaps for operators in a reasonable class, which lead eventually to the weak unique continuation property for differential inequalities with optimal conditions on potentials. We also get some Carleman inequalities for general operators with simple or double characteristics. The gaps here are not as good as in the first case. But we may prove the gaps in these inequalities are sharp in general. Actually we will provide counterexamples to prove such gaps are sharp in Carleman inequalities for operators in some subclasses of simple or double characteristics class. In particular, we prove that there is no Carleman inequality with positive gap for the highest order term for any operator whose symbol has double characteristics.