Carleman estimates and unique continuation for solutions to homogeneous linear PDE
Matthias Eller
Abstract:
Holmgren's Theorem states that solutions to linear partial differential equation with analytic coefficients satisfy the unique continuation property across non-characteristic surfaces. For operators with non-analytic coefficients Carleman estimates can be used to establish unique continuation across strongly pseudo-convex surfaces. An introduction to Carleman estimates for second order operators will be presented. The focus is on classical results by Hormander (1963) which apply to elliptic and hyperbolic operators. Isakov (1980) established Carleman estimates for parabolic and Schrodinger operators. Originally Carleman estimates were proved for compactly supported functions. Tataru (1996) gave a general approach to Carleman estimates without requiring compact support. Carleman estimates for solutions to boundary value problems have become valuable tools in boundary control problems and coefficient identification problems.