Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Initial-boundary value problems for hyperbolic systems in regions with corners. I


Author: Stanley Osher
Journal: Trans. Amer. Math. Soc. 176 (1973), 141-164
MSC: Primary 35L50
DOI: https://doi.org/10.1090/S0002-9947-1973-0320539-5
MathSciNet review: 0320539
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In recent papers Kreiss and others have shown that initial-boundary value problems for strictly hyperbolic systems in regions with smooth boundaries are well-posed under uniform Lopatinskiĭ conditions. In the present paper the author obtains new conditions which are necessary for existence and sufficient for uniqueness and for certain energy estimates to be valid for such equations in regions with corners. The key tool is the construction of a symmetrizer which satisfies an operator valued differential equation.


References [Enhancements On Off] (What's this?)

  • [1] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 25 #5186. MR 0141789 (25:5186)
  • [2] R. G. Douglas and R. Howe, On the $ {C^\ast}$-algebra of Toeplitz operators on the quarter plane, Trans. Amer. Math. Soc. 158 (1971), 203-217. MR 0288591 (44:5787)
  • [3] R. Hersh, Mixed problems in several variables, J. Math. Mech. 12, (1963), 317-334. MR 26 #5304. MR 0147790 (26:5304)
  • [4] V. A. Kondrat'ev, Boundary-value problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1963), 17-29 = Soviet Math. Dokl. 4 (1963), 1600-1602. MR 28 #1383. MR 0158157 (28:1383)
  • [5] L. Kraus and L. Levine, Diffraction by an elliptic cone, Comm. Pure Appl. Math. 14 (1961), 49-68. MR 22 #10554. MR 0119794 (22:10554)
  • [6] H.-O. Kreiss, Initial boundary valued problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298. MR 0437941 (55:10862)
  • [7] I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimension corner, Comm. Pure Appl. Math. 24 (1971), 381-393. MR 0412616 (54:738)
  • [8] Ja. B. Lopatinskiĭ, On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations, Ukrain. Mat. Ž. 5 (1953), 123-151. MR 17, 494. MR 0073828 (17:494b)
  • [9] A. S. Peters, Water waves over sloping beaches and the solution of a mixed boundary value problem for $ {{\mathbf{\Delta }}^2}\phi - {K^2}\phi - 0$ in a sector, Comm. Pure Appl. Math. 5 (1952), 87-108. MR 13, 789. MR 0046807 (13:789e)
  • [10] J. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math. 24 (1971). MR 0606239 (58:29326)
  • [11] J. Rauch, $ {\mathcal{L}_2}$ is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math. 25 (1972), 265-285. MR 0298232 (45:7284)
  • [12] R. Sakomoto, Mixed problems for hyperbolic equations. I, II, J. Math. Kyoto Univ. 10 (1970), 349-373, 403-417.
  • [13] L. Sarason, On weak and strong solutions of boundary value problems, Comm. Pure Appl. Math. 15 (1962), 237-288. MR 27 #460. MR 0150462 (27:460)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L50

Retrieve articles in all journals with MSC: 35L50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0320539-5
Keywords: Hyperbolic equations, initial boundary conditions, symmetrizer, energy estimate, well-posedness
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society