References
Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15, 119–147 (1962).
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).
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).
Arima, R., On general boundary value problems for parabolic equations. J. Math. Kyoto Univ. 4, 207–243 (1964).
Atiyah, M. F., Global aspects of the theory of elliptic differential operators. Proc. International Congress of Mathematicians, Moscow 1966, 7–14.
Bergendal, G., Convergence and summability of eigenfunction expansions connected with elliptic differential operators. Medd. Lunds Univ. Mat. Sem. 15, 1–63 (1959).
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).
Clark, C., The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems. SIAM Review 9, 627–646 (1967).
Dunford, N., & J. T. Schwartz, Linear Operators, Part II. New York: Interscience 1963.
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).
Eidelman, S. D., Parabolic Systems. “Nauka”, Moscow 1964 (Russian).
Friedman, A., Partial Differential Equations of Parabolic Type. Englewood Cliffs, N.J. Prentice-Hall 1964.
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.
Greiner, P., An Asymptotic Expansion for the Heat Equation. Proc. Sym. Pure Math. 16, 133–135 (1970).
Hille, E., & R. S. Phillips, Functional analysis and semigroups. AMS Coll. Publ. 31 (1957).
Hörmander, L., On the theory of general partial differential operators. Acta Math. 94, 161–248 (1955).
Hörmander, L., Linear Partial Differential Operators. Berlin-Göttingen-Heidelberg: Springer 1963.
Hörmander, L., Pseudo-differential operators. Comm. Pure Appl. Math. 18, 501–517 (1964).
Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. (1) 83, 129–209 (1966).
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).
Hörmander, L., On the Atiyah-Bott-Lefschetz fixed point formula. Unpublished manuscript.
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).
Kac, M., Can one hear the shape of a drum? Amer. Math. Monthly 73, 1–23 (1966).
Kohn, J. J., & L. Nirenberg, An algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18, 269–305 (1965).
Kotake, T., & M. S. Narasimhan, Regularity theorems for fractional powers of linear elliptic operators. Bull. Soc. Mat. de France 90, 449–472 (1962).
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.
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).
McKean, H. P., Jr., & I. M. Singer, Curvature and the eigenvalues of the Laplacian. J. Diff. Geometry 1, 43–69 (1967).
Minakshisundaram, S., A generalization of Epstein zeta functions. Can. J. Math. 1, 320–327 (1949).
Minakshisundaram, S., & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Ibid. 1, 242–256 (1949).
Mizohata, S., Sur les propriétés asymptotiques des valeurs propres pour les opérateurs elliptiques. J. Math. Kyoto Univ. 4, 399–428 (1965).
Palais, R., et al., Seminar on the Atiyah-Singer index theorem. Ann. Math. Studies, 1965, Study 57.
Pleijel, A., A study of certain Green's functions with applications in the theory of vibrating membranes. Ark. Mat. 2, 553–569 (1954).
Seeley, R. T., Integro-differential operators on vector bundles. Trans. Amer. Math. Soc. 117, 167–204 (1965).
Seeley, R. T., Singular integrals and boundary value problems. Amer. J. Math. 88, 781–809 (1966).
Seeley, R. T., The powers A s of an elliptic differential operator. Proc. Sym. Pure Math. 10, 288–307 (1968).
Solonnikov, V. A., On the boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov 83 (1965).
Steenrod, N., The Topology of Fibre Bundles. Princeton, 1951.
Author information
Authors and Affiliations
Additional information
Communicated by M. Kac
This research was partially supported by the National Research Council of Canada under Grant A-3017. The results were announced in [14].
Rights and permissions
About this article
Cite this article
Greiner, P. An asymptotic expansion for the heat equation. Arch. Rational Mech. Anal. 41, 163–218 (1971). https://doi.org/10.1007/BF00276190
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00276190