Abstract
This chapter is a summary of the basic definitions and results from the theory of distributions or generalized functions which will be used in subsequent chapters. Distribution theory has become a convenient tool in the study of partial differential equations. Many problems in partial differential equations can be formulated in terms of abstract operators acting between suitable spaces of distributions, and these operators are then analyzed by the methods of functional analysis. The virtue of this approach is that a given problem is stripped of extraneous data, so that the analytic core of the problem is revealed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Global Analysis Pure and Applied: Series B, vol. 2. Addison-Wesley, Reading (1983)
Aronszajn, N., Smith, K.T.: Theory of Bessel potentials I. Ann. Inst. Fourier (Grenoble), 11, 385–475 (1961)
Chazarain, J., Piriou, A.: Introduction à la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)
de Rham, G.: Variétés différentiables. Hermann, Paris (1955)
Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions I. Properties and Operations. Academic, New York/London (1964)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators. Reprint of the 1994 edition, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/Heidelberg/New York/Tokyo (2007)
Lang, S.: Differential Manifolds. Addison-Wesley, Reading (1972)
Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, vol. 51. American Mathematical Society, Providence (1997)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Peetre, J.: Réctification à l’article “Une caractérisation abstraite des opérateurs différentiels”. Math. Scand. 8, 116–120 (1960)
Schwartz, L.: Théorie des distributions. Nouvelle édition, Hermann, Paris (1966)
Stein, E.M.: The differentiability of functions in R n. Ann. Math. 113, 383–385 (1981)
Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York/London (1967)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1995). Reprint of the second edition (1944)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Taira, K. (2014). Theory of Distributions. In: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43696-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-43696-7_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-43695-0
Online ISBN: 978-3-662-43696-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)