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Mixed Problems for Hyperbolic Equations of General Order

Published online by Cambridge University Press:  20 November 2018

G. F. D. Duff
G. F. D. Duff*
Affiliation:
University of Toronto
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The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.

For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 195 - 221
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Campbell, L. L. and Robinson, A., Mixed problems for hyperbolic differential equations, Proc. Lond. Math. Soc. (3), 5 (1955), 129-47.Google Scholar
2. D. Duff, G. F., A mixed problem for normal hyperbolic linear partial differential equations of second order, Can. J. Math., 9 (1957), 141-60.Google Scholar
3. D. Duff, G. F., Mixed problems for linear systems, Can. J. Math., 10 (1958), 127-60.Google Scholar
4. G.|årding, L., Dirichlefs problem, Math. Scand., 1 (1953), 55–72.Google Scholar
5. G.|årding, L., Solution directe du problème de Cauchy pour les équations hyperboliques, Proc. Coll. Int. du C.N.R.S. LXXI (Paris, 1956), 71–90.Google Scholar
6. H, L.örmander, On the theory of general partial differential operators, Acta Math., 94 (1955), 161–248.Google Scholar
7. Kryzyanski, M. and Schauder, J., Quasi-lineare Differentialgleichungen zweiter Ordnung vom hyperbolischen Typus, Gemischte Randwertaufgeben, Studia Math., 6 (1936), 152-89.Google Scholar
8. Kuipers, L. and Meulenbeld, B., Symmetric polynomials with non-negative coefficients, Proc. Amer. Math. Soc, 6 (1955), 88–93.Google Scholar
9. Leray, J., Hyperbolic differential equations (Princeton, 1953).Google Scholar
10. Lions, J. L., Problèmes aux limites en théorie des distributions, Acta Math., 94 (1955), 13–153.Google Scholar
11. Littlewood, D. E., The theory of group characters (Cambridge, 1939).Google Scholar
12. Thomée, V., Estimates of the Friedrichs-Lewy type for mixed problems in the theory of linear hyperbolic differential equations in two independent variables, Math. Scand., 5 (1957), 93-113.Google Scholar