On Carleman and observability estimates for wave equations on time-dependent domains. (English) Zbl 1437.35441
The author establishes new Carleman estimates for the wave equation using a geometric approach. It is noted that the main features of the derived observability inequalities are that they apply to a fully general class of time-dependent domains, with timelike moving boundaries, they apply to linear wave equations in any spatial dimension and with general time-dependent lower order coefficients, and they allow for smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. The paper is rather extensive, its structure is as follows. First, the author defines the objects and notations to be used throughout the paper. Then, the main Careleman estimate is stated and proved. In the process, the author introduces the “warped” metric and establishes its basic properties. The next part of the paper is dedicated to the precise statements of the main observability inequalities. Finally, in the conclusive part, the author discusses some consequencess of the main observability results.
Reviewer: Vyacheslav I. Maksimov (Yekaterinburg)
MSC:
| 35L05 | Wave equation |
| 93B05 | Controllability |
| 93B07 | Observability |
| 93B27 | Geometric methods |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
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