We will discuss weighted $L^2$-estimates of solutions of general partial differential equations of second order. We introduce the so-called pseudo-convexity condition for the weight function and give examples of such functions for elliptic and hyperbolic operators. Then we formulate Carleman estimates with boundary terms, and give an elementary proof for a particular case of the Helmholtz operator. This proof illustrates the general case and gives new estimates with constants not depending on the wave number.

Uniqueness and stability in the Cauchy problem

Here, following the classical Carleman idea, we apply Carleman estimates to derive uniqueness results and stability estimates of the continuation of solutions to partial differential equations. We give the counterexample of Fritz John which shows importance of pseudo-convexity and outline recent progress in increased stability for the Helmholtz equation.