Mathematical Research Letters
Volume 12 (2005)
Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds
Pages: 37 – 47
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 $(0,T)\times\Omega$). 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.