Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T13:58:03.045Z Has data issue: false hasContentIssue false

A Mixed Problem for Normal Hyperbolic Linear Partial Differential Equations of Second Order

Published online by Cambridge University Press:  20 November 2018

G. F. D. Duff*
Affiliation:
University of Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the theory of hyperbolic differential equations a mixed boundary value problem involves two types of auxiliary conditions which may be described as initial and boundary conditions respectively. The problem of Cauchy, in which only initial conditions are present, has been studied in great detail, starting with the early work of Riemann and Volterra, and the well-known monograph of Hadamard (4). A modern treatment of great generality has been given by Leray (7).

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Banach, S. and Saks, S., Sur le convergence forte dans les champs Lp , Studia Math., 2 (1930), 5157.Google Scholar
2. Duff, G. F. D., Uniqueness in boundary value problems for the second order hyperbolic equation, Can. J. Math., 8 (1956), 8696.Google Scholar
3. Friedrichs, K. and Lewy, H., Ueber die Eindeutigkeit und das Abhängigkeitsgebeit der Lösungen beim Anfangswenproblem linear hyperbolischer Differentialgleichungen, Math. Ann., 98 (1928), 192204.Google Scholar
4. Hadamard, J., Lectures on Cauchy's problem in linear partial differential equations (New York, 1952).Google Scholar
5. Krzyzanski, M. and Schauder, J., Quastlineare Differentialgletchungen zweiter Ordnung vom hyperbolischen Typus: Gemischte Randwertaufgaben, Studia Math., 6 (1936), 162189.Google Scholar
6. Ladyshenskaya, O., Smesannaya zadaca dlya gyperboliceskova uravneniya (Moscow, 1953).Google Scholar
7. Leray, J., Hyperbolic Differential Equations (Princeton, 1953).Google Scholar
8. Robinson, A. and Campbell, L., Mixed problems for hyperbolic partial differential equations, Proc. London Math. Soc. (3), 5 (1955), 129147.Google Scholar
9. Schauder, J., Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung …, Fund. Math., 24 (1935), 213246.Google Scholar
10. Schauder, J., Gemischte Randwertaufgaben bei Pamellen Differentialgleichungen vom hyperbolischen Typus, Studia Math., 6 (1936), 190198.Google Scholar