Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds

Abstract

This article concerns some quantitative versions of unique continuation known as observability inequalities. One of them is a lower bound on the spectral projectors of the Dirichlet Laplacian which generalizes the unique continuation of an eigenfunction from any open set Omega. Another one is equivalent to the interior null-controllability in time T of the heat equation with Dirichlet condition (the input function is a source in (0,T) x Omega). On a compact Riemannian manifolds, these inequalities are known to hold for arbitrary T and Omega. This article states and links these observability inequalities on a complete non-compact Riemannian manifold, and tackles the quite open problem of finding which Omega and T ensure their validity. It proves that it is sufficient for Omega to be the exterior of a compact set (for arbitrary T), but also illustrates that this is not necessary. It provides a necessary condition saying that there is no sequence of balls going infinitely far away from Omega without shrinking in a generalized sense (depending on T) which also applies when the distance to Omega is bounded.