Skip to main content
SpringerLink
Log in

Geometrical optics and the corner problem

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Agmon, S., Problèmes mixtes pour les équations hyperboliques d'ordre supéreure. Colloques Internationales du Centre National de la Récherche Scientifique, 117, Paris 1962.

  2. Agemi, R., On energy inequalities of mixed problems for hyperbolic equations of second order. J. Fac. Science, Hokkaido Univ. (1) 21, 221–236 (1971).

    Google Scholar 

  3. Agranovich, M. S., Theorem on matrices depending on parameters and its applications to hyperbolic systems. Funktstional'nyi Analiz i ego Prilozheniya 6, 1–11 (1972).

    Google Scholar 

  4. Bloom, C. O., On the validity of the geometrical theory of diffraction by starshaped cylinders. J. Math. Anal. and Appl. 40, 107–121 (1972).

    Google Scholar 

  5. Borovikov, V. A., The elementary solution of partial differential equations with constant coefficients. Trudy Moskov. Mat. Obšč. 68, 159–257 (1959).

    Google Scholar 

  6. Burridge, R., Lacunas in two-dimensional wave propagation. Proc. Cambridge Phil. Soc. 63, 819–825 (1967).

    Google Scholar 

  7. Courant, R., & D. Hilbert, Methods of Mathematical Physics, Vol. II, Partial Differential Equations. New York: Interscience Publishers 1962.

    Google Scholar 

  8. Duff, G. F. D., On wave fronts and boundary waves. Comm. Pure Appl. Math. 17, 189–225 (1964).

    Google Scholar 

  9. Duistermaat, J. J., & L. Hörmander, Fourier integral operators II. Acta Math. 128, 183–269 (1972).

    Google Scholar 

  10. Gärding, L., Solution directe du problème de Cauchy pour les équations hyperboliques. Proc. Coll. Int. du C.N.R.S. 71, 71–90 (1956).

    Google Scholar 

  11. Hersh, R., Mixed problems in several variables. J. Math. Mech. 12, 317–334 (1963).

    Google Scholar 

  12. Hörmander, L., Fourier integral operators, I. Acta Mathematica 127, 79–183 (1971).

    Google Scholar 

  13. Ikawa, M., Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20, 580–608 (1968).

    Google Scholar 

  14. Keller, J. B., Geometrical theory of diffraction. J. Opt. Soc. Amer. 52, 116–130 (1962).

    Google Scholar 

  15. Kupka, I. A. K., & S. Osher, On the wave equation in a multi-dimensional corner. Comm. Pure Appl. Math. 24, 381–394 (1971).

    Google Scholar 

  16. Kajitani, K., Initial-boundary value problems for first order hyperbolic systems. R.I.M.S., Kyoto Univ. 7, 181–204 (1971/72).

    Google Scholar 

  17. Kreiss, H.-O., Initial boundary value problems for first order hyperbolic systems. Comm. Pure Appl. Math. 23, 277–298 (1970).

    Google Scholar 

  18. Lax, P. D., Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24, 627–646 (1957).

    Google Scholar 

  19. Lax, P. D., & R. S. Phillips, Scattering Theory. New York: Academic Press 1967.

    Google Scholar 

  20. Ludwig, D., Exact and asymptotic solutions of the Cauchy problem. Comm. Pure Appl. Math. 13, 473–508 (1960).

    Google Scholar 

  21. Ludwig, D., & C. S. Morawetz, An inequality for the reduced wave operator and the justification of geometrical optics. Comm. Pure Appl. Math. 21, 187–203 (1968).

    Google Scholar 

  22. Nirenberg, L., Lectures on Linear Partial Differential Equations, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics # 17.

  23. Osher, S., Initial-boundary value problems for hyperbolic systems in regions with corners, I. Trans. Amer. Math. Soc. 176, 141–164 (1973).

    Google Scholar 

  24. Osher, S., An ill posed problem for a hyperbolic equation near a corner. Bull. Amer. Math. Soc. 79, 1043–1044 (1973).

    Google Scholar 

  25. Ralston, J. V., Note on a paper of Kreiss. Comm. Pure Appl. Math. 24, 759–762 (1971).

    Google Scholar 

  26. Ralston, J. V., Solutions of the wave equation with localized energy. Comm. Pure Appl. Math. 22, 807–823 (1969).

    Google Scholar 

  27. Rauch, J., ℒ2 is a continuable initial condition for Kreiss' mixed problem. Comm. Pure Appl. Math. 25, 265–285 (1972).

    Google Scholar 

  28. Rauch, J., Energy and resolvent inequalities for hyperbolic mixed problems. J. Diff. Equations 11, 528–540 (1972).

    Google Scholar 

  29. Rauch, J., & F. J. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189, 303–318 (1974).

    Google Scholar 

  30. Sadamatsu, T., On mixed problems for hyperbolic systems of first order with constant coefficients. J. Math. of Kyoto Univ. 9, 339–361 (1969).

    Google Scholar 

  31. Sakamoto, R., Mixed problems for hyperbolic equations, II. J. Math, of Kyoto Univ. 10, 403–417 (1970).

    Google Scholar 

  32. Sarason, L., On hyperbolic mixed problems. Arch. Rational Mech. Anal. 18, 311–334 (1965).

    Google Scholar 

  33. Sarason, L., Elliptic regularization for symmetric positive systems. J. Math. Mech. 16, 807–827 (1967).

    Google Scholar 

  34. Sarason, L., Symmetrizable systems in regions with corners and edges. J. Math. Mech. 19, 601–607 (1970).

    Google Scholar 

  35. Sarason, L., On weak and strong solutions of boundary value problems. Comm. Pure Appl. Math., 15, 237–288 (1962).

    Google Scholar 

  36. Shirota, T., & K. Asano, On mixed problems for regularly hyperbolic systems. J. of the Faculty of Science, Kokkaido Univ. (1) 21, 1–45 (1970).

    Google Scholar 

  37. Shirota, T., & R. Agemi, On certain mixed problems for hyperbolic equations of higher order, III. Proc. Japan Acad. 45, 854–858 (1969).

    Google Scholar 

  38. Shirota, T., & R. Agemi, On necessary and sufficient conditions for L 2-well posedness of mixed problems for hyperbolic equations. J. Faculty of Science, Hokkaido Univ. (1) 21, 133–151 (1970).

    Google Scholar 

  39. Strang, G., On strong hyperbolicity. J. Math. Kyoto Univ. 6, 397–417 (1967).

    Google Scholar 

  40. Strang, G., Hyperbolic initial-boundary value problems in two unknown. J. Diff. Equations 6, 161–171 (1969).

    Google Scholar 

  41. Atiyah, M. F., R. Bott, & L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients. Acta Math. 126, 109–189 (1970).

    Google Scholar 

  42. Petrovsky, I. G., On the diffusion of waves and the lacunas for hyperbolic equations. Mat. 86, 17(59), 289–370 (1945).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle

    L. Sarason & J. A. Smoller

  2. Department of Mathematics, University of Michigan, Ann Arbor

    L. Sarason & J. A. Smoller

Authors
  1. L. Sarason
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Smoller
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by G. Strang

Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462, at Mathematics Research Center, University of Wisconsin, Madison.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarason, L., Smoller, J.A. Geometrical optics and the corner problem. Arch. Rational Mech. Anal. 56, 34–69 (1974). https://doi.org/10.1007/BF00279820

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00279820

Keywords

Search

Navigation