Contour integrals for the ultrahyperbolic wave equation
Abstract
The Penrose transform for the ultrahyperbolic wave equation is related to the problem of specifying data on an initial spacelike 2-plane. It is shown that a smooth solution can always be represented locally by a twistor contour integral formula, and this fact is used to rederive Asgeirsson’s mean value theorem, as well as the classic result that it is not possible to specify arbitrary ‘initial data’ on a non-null 3-plane. The global problem of constructing a twistor transform in the compactified space-time with signature + + - - is also considered.
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Woodhouse (1992) studied the case of two space and two time dimensions with initial data on a spacelike hypersurface of codimension 2, using the Penrose twistor transform in the real case, and recovered the uniqueness result of Courant and its implicit constraints on well-posed initial data for the Cauchy problem.
“In related and prior work, Woodhouse (1992) studied the case of two space and two time dimensions with initial data on a spacelike hypersurface (thus of codimension 2), using the Penrose twistor transform in the real case. He also recovered the uniqueness result of Courant and its implicit constraints on well-posed inital data for the Cauchy problem.”
From On determinism and well-posedness in multiple time dimensions (2009) · cites this paper as Woodhouse (1992)
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Penrose, R., & Rindler, W. (1984) · doi:10.1017/cbo9780511564048 · Cited by 2,172 · see citing works
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On determinism and well-posedness in multiple time dimensions
Craig, W., & Weinstein, S. (2009) · Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rspa.2009.0097 · Cited by 62 · see citing works
- Authors
- Woodhouse, N. M. J.
- Affiliations
- University of Oxford (GB)
- Corresponding author
- Woodhouse, N. M. J.
- Venue
- Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- Publisher
- The Royal Society
- Published
- 1992-07-08
- Volume
- 438
- Issue
- 1902
- Pages
- 197-206
- ISSN
- 0962-8444 (ISSN-L)
- 2053-9177
- OpenAlex
- W1987280006
- Cited by
- 22
- References
- 1
- Citations by year
- 2024: 1 · 2018: 1 · 2017: 2 · 2016: 1 · 2014: 2 · 2012: 1
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Differential Equations and Boundary Problems
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical and Theoretical Analysis
- Keywords
- Penrose transform, twistor theory, contour integrals, initial data, signature
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