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Archive for Rational Mechanics and Analysis

, Volume 41, Issue 3, pp 163–218 | Cite as

An asymptotic expansion for the heat equation

  • Peter Greiner
Article

Keywords

Neural Network Complex System Nonlinear Dynamics Asymptotic Expansion Electromagnetism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15, 119–147 (1962).Google Scholar
  2. 2.
    Agmon, S., A. Douglis, & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Comm. Pure Appl. Math. 12, 623–727 (1959); II. ibid. 17, 35–92 (1964).Google Scholar
  3. 3.
    Agranovich, M. S., & M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type. Uspehi Mat. Nauk 19, no. 3 (117), 53–161 (1964). Russian Math. Surveys 19, no. 3, 53–157 (1964).Google Scholar
  4. 4.
    Arima, R., On general boundary value problems for parabolic equations. J. Math. Kyoto Univ. 4, 207–243 (1964).Google Scholar
  5. 5.
    Atiyah, M. F., Global aspects of the theory of elliptic differential operators. Proc. International Congress of Mathematicians, Moscow 1966, 7–14.Google Scholar
  6. 6.
    Bergendal, G., Convergence and summability of eigenfunction expansions connected with elliptic differential operators. Medd. Lunds Univ. Mat. Sem. 15, 1–63 (1959).Google Scholar
  7. 7.
    Boutet de Monvel, L., Comportement d'un opérateur pseudo-différentiel sur un variété à bord. II. Pseudo-noyaux de Poisson. J. d'Analyse Mat. 17, 255–304 (1966).Google Scholar
  8. 8.
    Clark, C., The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems. SIAM Review 9, 627–646 (1967).Google Scholar
  9. 9.
    Dunford, N., & J. T. Schwartz, Linear Operators, Part II. New York: Interscience 1963.Google Scholar
  10. 10.
    Eidelman, S. D., On fundamental solutions of parabolic systems, I. Mat. Sb. 38 (80), 51–92 (1956); II. ibid. 53 (95), 73–136 (1961); Amer. Math. Soc. Transl. (2) 41, 1–48, 49–120 (1964).Google Scholar
  11. 11.
    Eidelman, S. D., Parabolic Systems. “Nauka”, Moscow 1964 (Russian).Google Scholar
  12. 12.
    Friedman, A., Partial Differential Equations of Parabolic Type. Englewood Cliffs, N.J. Prentice-Hall 1964.Google Scholar
  13. 13.
    Gelfand, I. M., & G. E. Shilov, Generalized Functions. v. II and III. Fizmatgiz, Moscow 1958. English translation, New York: Academic Press. v. II, 1968, v. III, 1967.Google Scholar
  14. 14.
    Greiner, P., An Asymptotic Expansion for the Heat Equation. Proc. Sym. Pure Math. 16, 133–135 (1970).Google Scholar
  15. 15.
    Hille, E., & R. S. Phillips, Functional analysis and semigroups. AMS Coll. Publ. 31 (1957).Google Scholar
  16. 16.
    Hörmander, L., On the theory of general partial differential operators. Acta Math. 94, 161–248 (1955).Google Scholar
  17. 17.
    Hörmander, L., Linear Partial Differential Operators. Berlin-Göttingen-Heidelberg: Springer 1963.Google Scholar
  18. 18.
    Hörmander, L., Pseudo-differential operators. Comm. Pure Appl. Math. 18, 501–517 (1964).Google Scholar
  19. 19.
    Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. (1) 83, 129–209 (1966).Google Scholar
  20. 20.
    Hörmander, L., On the Riesz Means of Spectral Functions and Eigenfunction Expansions for Elliptic Differential Operators. Lecture at the Belfer Graduate School, Yeshiva University, Nov. 16, 1966 (Mimeographed).Google Scholar
  21. 21.
    Hörmander, L., On the Atiyah-Bott-Lefschetz fixed point formula. Unpublished manuscript.Google Scholar
  22. 22.
    Ito, S., The fundamental solution of the parabolic equation in a differentiable manifold, I. Osaka Math. J. 5, 75–92 (1953); II. ibid. 6, 167–185 (1954).Google Scholar
  23. 23.
    Kac, M., Can one hear the shape of a drum? Amer. Math. Monthly 73, 1–23 (1966).Google Scholar
  24. 24.
    Kohn, J. J., & L. Nirenberg, An algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18, 269–305 (1965).Google Scholar
  25. 25.
    Kotake, T., & M. S. Narasimhan, Regularity theorems for fractional powers of linear elliptic operators. Bull. Soc. Mat. de France 90, 449–472 (1962).Google Scholar
  26. 26.
    Ladyzenskaja, O. A., V. A. Solonnikov, & N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. “Nauka”, Moscow: Fizmatgiz 1967. English transl. AMS, Providence, 1968.Google Scholar
  27. 27.
    Lopatinskii, Ja. B., On a method for reducing boundary problems for systems of differential equations of elliptic type to regular integral equations. Ukrain. Mat. Z. 5, 123–151 (1953).Google Scholar
  28. 28.
    McKean, H. P., Jr., & I. M. Singer, Curvature and the eigenvalues of the Laplacian. J. Diff. Geometry 1, 43–69 (1967).Google Scholar
  29. 29.
    Minakshisundaram, S., A generalization of Epstein zeta functions. Can. J. Math. 1, 320–327 (1949).Google Scholar
  30. 30.
    Minakshisundaram, S., & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Ibid. 1, 242–256 (1949).Google Scholar
  31. 31.
    Mizohata, S., Sur les propriétés asymptotiques des valeurs propres pour les opérateurs elliptiques. J. Math. Kyoto Univ. 4, 399–428 (1965).Google Scholar
  32. 32.
    Palais, R., et al., Seminar on the Atiyah-Singer index theorem. Ann. Math. Studies, 1965, Study 57.Google Scholar
  33. 33.
    Pleijel, A., A study of certain Green's functions with applications in the theory of vibrating membranes. Ark. Mat. 2, 553–569 (1954).Google Scholar
  34. 34.
    Seeley, R. T., Integro-differential operators on vector bundles. Trans. Amer. Math. Soc. 117, 167–204 (1965).Google Scholar
  35. 35.
    Seeley, R. T., Singular integrals and boundary value problems. Amer. J. Math. 88, 781–809 (1966).Google Scholar
  36. 36.
    Seeley, R. T., The powers A s of an elliptic differential operator. Proc. Sym. Pure Math. 10, 288–307 (1968).Google Scholar
  37. 37.
    Solonnikov, V. A., On the boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov 83 (1965).Google Scholar
  38. 38.
    Steenrod, N., The Topology of Fibre Bundles. Princeton, 1951.Google Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • Peter Greiner
    • 1
  1. 1.Department of MathematicsUniversity of TorontoCanada

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