Rocky Mountain Journal of Mathematics

Low regularity ray tracing for wave equations with Gaussian beams

Alden Waters

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Abstract

We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb {R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 3 (2019), 1005-1027.

Dates
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1563847245

Digital Object Identifier
doi:10.1216/RMJ-2019-49-3-1005

Mathematical Reviews number (MathSciNet)
MR3983312

Zentralblatt MATH identifier
07088348

Subjects
Primary: 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]
Secondary: 35R30: Inverse problems 35L20: Initial-boundary value problems for second-order hyperbolic equations 58J45: Hyperbolic equations [See also 35Lxx] 35A22: Transform methods (e.g. integral transforms)

Keywords
Control theory inverse problems wave equations observability low regularity coefficients Gaussian beams

Citation

Waters, Alden. Low regularity ray tracing for wave equations with Gaussian beams. Rocky Mountain J. Math. 49 (2019), no. 3, 1005--1027. doi:10.1216/RMJ-2019-49-3-1005. https://projecteuclid.org/euclid.rmjm/1563847245


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