Local properties of solutions of elliptic partial differential equations

  • A. P. Calderón
  • A. Zygmund
Chapter
Part of the Mathematics and Its Applications book series (MAEE, volume 41)

Abstract

The purpose of this paper is to establish pointwise estimates for solutions of elliptic partial differential equations and systems. The results presented here differ from the familiar ones in that they give information about the behavior of solutions at individual points. More specifically, we obtain two kinds of results. On the one hand, we establish inequalities for solutions and their derivatives at isolated individual points. On the other, we also obtain results of the „almost everywhere“ type. Theorems 1 and 2 below summarize the main results.

Keywords

Compact Support Taylor Expansion Differentiable Function Inverse Fourier Transform Singular Integral Operator 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    N. Aronszajn and K. T. Smith, Theory of Bessel potentials, Part I, Studies in Eigenvalue Problems, Technical Report 22, University of Kansas, Department of Mathematics (1959), p. 1–113.Google Scholar
  2. [2]
    H. Bateman, Tables of integral transforms, New York 1954.Google Scholar
  3. [3]
    H. Bateman, Higher transcendental functions, New York 1953.Google Scholar
  4. [4]
    A. P. Calderón, On the differentiability of absolutely continuous functions, Kivista de Math. Univ. Parma 2 (1951), p. 203–212.MATHGoogle Scholar
  5. [5]
    H. Bateman, Singular integrals, Course notes of lectures given at the M. I. T., 1958–59.Google Scholar
  6. [6]
    A. P. Calderón and A. Zygmund, On singular integrals, Amer. Jour. of Math. 78. 2 (1956), p. 290–309.Google Scholar
  7. [7]
    A. P. Calderón, Singular integral operators and differential equations, Amer. Jour. of Math. 79 (1957), p. 901–921.MATHCrossRefGoogle Scholar
  8. [8]
    L. Cesàri, Sulle funzioni assolutamente continue in due variabili, Annali di Pisa 10 (1041), p. 91–101.Google Scholar
  9. [9]
    J. Horváth, Sur les fonctions conjuguées à plusieurs variables, Indagationes Math. (1953), p. 17–29.Google Scholar
  10. [10]
    J. Marcinkiewicz, Sur les séries de Fourier, Fund. Math. 27 (1930), p. 38–69.Google Scholar
  11. [11]
    L. Nirenberg, On elliptic partial differential equations. Annali della Scuola Norm. Sup. Pisa 13 (1959), p. 116–162.MathSciNetGoogle Scholar
  12. [12a]
    W. H. Oliver, Differential properties of real functions, Ph. D. Dissertation, Univ. of Chicago, 1951, p. 1–109;Google Scholar
  13. [12b]
    W. H. Oliver, An existence theorem for the n-th Peano differential, Abstract, Bull. Amer. Math. Soc. 57 (1951), p. 472.Google Scholar
  14. [13]
    G. H. Watson, A treatise on the theory of Bessel functions, Cambridge 1944.MATHGoogle Scholar
  15. [14]
    H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), p. 63–89.MathSciNetCrossRefGoogle Scholar
  16. [15]
    A. Zygmund, Trigonometric Series, vols I and II, Cambridge 1959.Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • A. P. Calderón
    • 1
  • A. Zygmund
    • 1
  1. 1.ChicagoUSA

Personalised recommendations