Journal
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
Volume 23, Issue 10, Pages 3459-3495Publisher
EUROPEAN MATHEMATICAL SOC-EMS
DOI: 10.4171/JEMS/1105
Keywords
Wave equation; inverse-square potential; Carleman estimates; weighted estimates
Categories
Funding
- ERC [862342, 801867]
- ICMAT-Severo Ochoa grant [CEX2019-000904-S]
- EPSRC [EP/R011982/1]
- European Research Council (ERC) [862342, 801867] Funding Source: European Research Council (ERC)
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Researchers established a new family of Carleman inequalities capturing natural boundary conditions and H-1 energy, and applied these estimates to prove a boundary observability property for the associated wave equations.
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.
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