A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

W. Jurkat; D. Nonnenmacher

Fundamenta Mathematicae (1994)

  • Volume: 146, Issue: 1, page 69-84
  • ISSN: 0016-2736

Abstract

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in ℝ n , we are led to an integration process over quite general sets A ⊆ q ℝ n with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

How to cite

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Jurkat, W., and Nonnenmacher, D.. "A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary." Fundamenta Mathematicae 146.1 (1994): 69-84. <http://eudml.org/doc/212052>.

@article{Jurkat1994,
abstract = {Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.},
author = {Jurkat, W., Nonnenmacher, D.},
journal = {Fundamenta Mathematicae},
keywords = {vector fields; non-absolutely convergent integrals; divergence theorem},
language = {eng},
number = {1},
pages = {69-84},
title = {A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary},
url = {http://eudml.org/doc/212052},
volume = {146},
year = {1994},
}

TY - JOUR
AU - Jurkat, W.
AU - Nonnenmacher, D.
TI - A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary
JO - Fundamenta Mathematicae
PY - 1994
VL - 146
IS - 1
SP - 69
EP - 84
AB - Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.
LA - eng
KW - vector fields; non-absolutely convergent integrals; divergence theorem
UR - http://eudml.org/doc/212052
ER -

References

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  1. [Fed] H. Federer, Geometric Measure Theory, Springer, New York, 1969. 
  2. [Jar-Ku 1] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits C 1 -transformations, Časopis Pěst. Mat. 109 (1984), 157-167. Zbl0555.26005
  3. [Jar-Ku 2] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (110) (1985), 116-139. Zbl0614.26007
  4. [Jar-Ku 3] J. Jarník and J. Kurzweil, A new and more powerful concept of the PU integral, ibid. 38 (113) (1988), 8-48. Zbl0669.26006
  5. [JKS] J. Jarník, J. Kurzweil and S. Schwabik, On Mawhin's approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), 356-380. Zbl0555.26004
  6. [Ju] W. B. Jurkat, The Divergence Theorem and Perron integration with exceptional sets, Czechoslovak Math. J. 43 (118) (1993), 27-45. Zbl0789.26005
  7. [Ju-No 1] W. B. Jurkat and D. J. F. Nonnenmacher, An axiomatic theory of non-absolutely convergent integrals in ℝ n , Fund. Math. 145 (1994), 221-242. Zbl0824.26007
  8. [Ju-No 2] W. B. Jurkat and D. J. F. Nonnenmacher, A generalized n-dimensional Riemann integral and the Divergence Theorem with singularities, Acta Sci. Math. (Szeged) 59 (1994), 241-256. Zbl0810.26007
  9. [Ju-No 3] W. B. Jurkat and D. J. F. Nonnenmacher, The Fundamental Theorem for the ν 1 -integral on more general sets and a corresponding Divergence Theorem with singularities, Czechoslovak Math. J., to appear. Zbl0832.26008
  10. [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, ibid. 31 (106) (1981), 614-632. Zbl0562.26004
  11. [No 1] D. J. F. Nonnenmacher, Sets of finite perimeter and the Gauss-Green Theorem with singularities, J. London Math. Soc., to appear. Zbl0835.26008
  12. [No 2] D. J. F. Nonnenmacher, A constructive definition of the n-dimensional ν(S)-integral in terms of Riemann sums, preprint 1992, to appear. 
  13. [Pf 1] W. F. Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), 665-685. Zbl0596.26007
  14. [Pf 2] W. F. Pfeffer, The Gauss-Green Theorem, Adv. in Math. 87 (1991), 93-147. Zbl0732.26013

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