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Some geometric aspects of hyperbolic boundary value problems


Authors: J. Brian Conrey and Michael W. Smiley
Journal: Proc. Amer. Math. Soc. 107 (1989), 591-601
MSC: Primary 35L20; Secondary 11H55, 35L70, 35P05, 47F05, 47H15
DOI: https://doi.org/10.1090/S0002-9939-1989-0975635-3
MathSciNet review: 975635
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ denote the linear operator associated with the wave equation when it is subjected to boundary conditions in both space and time. The properties of invertibility or partial invertibility of $ L$, and compactness of the (partial) inverse when it exists, are characterized in terms of the space time domain $ \Omega \times (0,T)$, for all rectangular domains $ \Omega \subset {{\mathbf{R}}^n}$.


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  • [1] Z. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [2] D. G. Bourgin and R. J. Duffin, The Dirichlet problem for the vibrating string equation, Bull. Amer. Math. Soc. 45(1939), 851-858. MR 0000729 (1:120f)
  • [3] H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. 8(3)(1983), 409-426. MR 693957 (84e:35010)
  • [4] F. E. Browder, A remark on the Dirichlet problem for non-elliptic self-adjoint partial differential operators, Rend. Circ. Mat. Palermo, 6(2)(1957), 249-253. MR 0104057 (21:2819)
  • [5] -, On the Dirichlet problem for linear non-elliptic partial differential equations II, Rend. Circ. Mat. Palermo, 7(2)(1958), 303-308. MR 0108650 (21:7365)
  • [6] D. R. Dunninger, The Dirichlet problem for a nonhomogeneous wave equation, Boll. Un. Math. Ital., 2-A(6)(1983), 253-265. MR 706662 (85g:35076)
  • [7] D. R. Dunninger and H. A. Levine, Uniqueness criteria for solutions to abstract boundary value problems, J. Differential Equations 22(6)(1976), 368-378. MR 0422795 (54:10781)
  • [8] D. W. Fox and C. Pucci, The Dirichlet problem for the wave equation, Ann. Mat. Pura Appl. 46(4)(1958), 155-182. MR 0104902 (21:3653)
  • [9] J. Hadamard, Equations aux derivees partielles, Enseign. Math. 35(1936), 5-42. MR 0220566 (36:3622c)
  • [10] T. Kato, Perturbation theory for linear operators, 2nd. ed., Springer-Verlag, New York, 1976. MR 0407617 (53:11389)
  • [11] I. S. Louhivaara, Über das Dirichletsche problem fur die selbstadjungierten linearen partiellen Differentialgleichungen zweiter Ordnung, Rend. Circ. Mat. Palermo 5(2)(1956), 260-274. MR 0086992 (19:281b)
  • [12] H. Lovicarova, Periodic solutions of a weakly nonlinear wave equation in one dimension, Czech. Math. J. 19(1969), 324-342. MR 0247249 (40:518)
  • [13] G. A. Margulis, Formes quadratrigues indefinies et flots unipotents sur les espaces homogènes, C. R. Acad. Sci. Paris Ser. I. Math. 304(1987), no. 10, 249-253. MR 882782 (88f:11027)
  • [14] -, Indefinite quadratic forms and unipotent flows on homogeneous spaces, (to appear in Semester on Dynamical Systems and Ergodic Theory, Warsaw 1986 Banach Center Publications).
  • [15] P. J. McKenna, On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational, Proc. Amer. Math. Soc. 93(1)(1985), 59-64. MR 766527 (86f:35017)
  • [16] L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, New York, 1977. MR 0249355 (40:2600)
  • [17] I. Niven and H. S. Zuckerman, The theory of numbers, 4th ed., Wiley, New York, 1980. MR 572268 (81g:10001)
  • [18] M. W. Smiley, Hyperbolic Boundary Value Problems. A Lax-Milgram approach and the vibrating string, Boll. Un. Mat. Ital. 5-A(6)(1986), 23-32. MR 833376 (87g:35131)
  • [19] -, Eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance, J. Math. Anal. Appl. 122(1)(1987), 129-151. MR 874965 (88c:47130)
  • [20] M. Vainberg, Variational methods for the study of nonlinear operators, (translated by A. Feinstein), Holden-Day, San Francisco, 1964. MR 0176364 (31:638)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0975635-3
Keywords: Wave equation, linear operator, spectrum, cluster point, quadratic form, Fredholm operator, compact operator
Article copyright: © Copyright 1989 American Mathematical Society

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