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Currents: The Deterministic Case

  • Vincenzo Capasso
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Currents are an extension of the concept of distributions, being themselves continuous linear functionals acting on a suitable space of differential forms; indeed, one might say that currents are differential forms having distributions as coefficients.

The first part of this chapter is devoted to a reminder of the main properties of distributions, as continuous linear functionals on the space of functions of compact support which are continuous with all possible derivatives. It is paid particular attention to the case of distributions associated with Radon measures.

Then the space of m-currents is introduced endowed with a suitable topology. Examples of distributions and currents are presented, with a particular attention to currents which anticipate the specific real applications presented in the relevant chapter.

Operations on currents and the definition of push-forward of a current are presented too.

References

  1. 6.
    Ash, R.: Real Analysis and Probability. Academic, London (1972)zbMATHGoogle Scholar
  2. 11.
    Bessaih, H., Coghi, M., Flandoli, F.: Mean field limit of interacting filaments and vector valued non linear PDEs. J. Stat. Phys. 166, 1276–1309 (2017)MathSciNetCrossRefGoogle Scholar
  3. 30.
    de Rham, G.: Differentiable Manifolds. Springer, Berlin (1984)CrossRefGoogle Scholar
  4. 36.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  5. 43.
    Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations I. Cartesian Currents. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 50.
    Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea, New York (1956)zbMATHGoogle Scholar
  7. 53.
    Laurent-Thiébaut, C.: Holomorphic Function Theory in Several Variables. An Introduction. Springer, London (2011)CrossRefGoogle Scholar
  8. 62.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  9. 68.
    Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow (1979)zbMATHGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  1. 1.ADAMSS (Centre for Advanced Applied Mathematical and Statistical Sciences)Universitá degli Studi di Milano La StataleMilanoItaly

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