Authors: Alberto Enciso (ICMAT-CSIC), Arick Shao and Bruno Vergara
Source: Journal of the European Mathematical Society vol. 23, no. 10
Date of publication: 9 June 2021
Carleman estimates are estimates for differential operators which involve exponential weights of arbitrarily high strength. Yet they are uniform with respect to the strength parameter, they are a powerful tool to establish quantitative uniqueness results for differential equations. Furthermore, they have found a wealth of applications including the study of inverse problems, observability and controllability properties, and the analysis of eigenvalues embedded in the continuous spectrum.
The objective in this paper is to derive Carleman estimates for wave operators with critically singular potentials, that is, with potentials that scale like the principal part of the operator. More specifically, the researchers are interested in the case of potentials that diverge as an inverse square on a convex hypersurface. For simplicity, they consider the simplest case where the aforementioned hypersurface is the unit sphere.
The key properties of the Carleman estimates they seek is that they are “sharp” and “global”. We say they are sharp, in that the weights that appear capture both the optimal decay rate of the solutions near the boundary (which is nontrivial and depends on the strength of the singular potential), as well as the natural energy that appears in the well-posedness theory for the equation. This property is not only desirable but also essential for applications such as boundary observability. By “global”, we mean that the Carleman estimate does not only involve the behavior of (suitably cut-off) functions near the hypersurface, but also capture their behavior inside the domain bounded by the hypersurface (in this case, the unit ball). This enables the authors to prove not only a unique continuation result, but also an interesting boundary observability property for the associated wave equations.
Without getting into unnecessary technicalities, the proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential.