Observability for wave equations with critically singular potentials

Abstract

In this article, we prove boundary observability estimates for wave equations on bounded domains $\Omega \subseteq \mathbb{R}^n$, with a critically singular potential that diverges as the inverse square distance to $\partial \Omega$. The result is applicable in all dimensions and for wave equations with general time-dependent lower order terms. The key geometric assumption is a convexity condition on $\Omega$. The main tool for our result is a global Carleman estimate that is carefully adapted to the critical singular nature of the potential.