Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

Alberto Enciso,Bruno Vergara,Arick Shao

doi:10.4171/jems/1105

Abstract

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains involving a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. 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. As an application of these estimates, we prove a boundary observability property for the associated wave equations.

Highlights

Our 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
The dispersive properties of wave equations with potentials that diverge as an inverse square at one point [7, 10] or an a hypersurface [4] have been thoroughly studied, as critically singular potentials are notoriously difficult to analyze
The methods employed in those references, which rely on the spectral analysis of a one-dimensional Bessel-type operator, provide very precise observability and controllability results. Another fruitful strategy for obtaining observability inequalities for a wide variety of PDEs is via Carleman-type estimates; see [36, 35] for some earliest applications, as well as [28, 41] for wave equations

Summary

Introduction

Our 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. As a result of this, Theorem 1.1 can be combined with standard arguments in order to prove the following rough statement: solutions to the wave equation with a critically singular potential on the boundary of a cylindrical domain satisfy boundary observability estimates, provided that the observation is made over a large enough timespan. If κ in Theorem 1.8 is replaced by (that is, we consider nonsingular wave equations), observability holds for any T > 1 This can be deduced from either the geometric control condition of [8] (see [9, 31]) or from standard Carleman estimates [5, 28, 41]. These generalize the classical Morawetz estimates to wave equations with critically singular potentials.

Preliminaries

Multiplier Inequalities

The Carleman Estimates

Observability