Carleman Type Estimates and Their Applications

Abstract

In these lecture notes we derive Carleman type estimates for second order linear partial differential operators and show some of their applications to the uniqueness in the Cauchy problem, approximate and exact controllability, and inverse problems. In section 2 we discuss pseudo-convexity and derive Carleman type estimates with boundary terms. In section 3 we obtain similar estimates with additional large parameter. Section 4 is devoted to uniqueness and stability in the Cauchy problem as well as applications to approximate and exact controllability. Section 5 exposes the Bukhgeim-Klibanov method of proving uniqueness in inverse problems by using Carleman estimates. Concluding section 6 exposes results on sharp uniqueness domains in the Cauchy problem, including recent fundamental theorem of Tataru and its generalizations. We will give complete proofs of some important results and will formulate some other interesting theorems referring for their proofs and other information either to the original papers or to the books [2], [12], [14], [18], [22].