Skip to main content
Log in

Stokes Formula for Banach Manifolds

  • Published:
Ukrainian Mathematical Journal Aims and scope

We propose a divergent version of the Stokes formula for a Banach manifold with uniform atlas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. V. Skorokhod, Integration in Hilbert Spaces [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  2. H-H. Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin (1975).

    Book  Google Scholar 

  3. A. V. Uglanov, Integration on Infinite-Dimensional Surfaces and Its Applications, Kluwer AP, Dordrecht (2000).

    Book  Google Scholar 

  4. Yu. V. Bogdanskii, “Banach manifolds with bounded structure and the Gauss–Ostrogradskii formula,” Ukr. Mat. Zh., 64, No. 10, 1299–1313 (2012); English translation: Ukr. Math. J., 64, No. 10, 1475–1494 (2013).

  5. Yu. V. Bogdanskii, “Divergence theorem in the L2-version. Application to the Dirichlet problem,” Ukr. Mat. Zh., 70, No. 5, 611–624 (2018); English translation: Ukr. Math. J., 70, No. 5, 702–718 (2018).

  6. N. V. Smorodina, “Gauss–Ostrogradskii formula for the configuration space,” Teor. Veroyatn. Primen., 35, No. 4, 727–739 (1990).

    MathSciNet  Google Scholar 

  7. D. L. Finkelshtein, Yu. G. Kondratiev, A. Yu. Konstantinov, and M. Rockner, “Gauss formula and symmetric extensions of Laplacian on configuration spaces,” Infinite-Dimensional Anal., Quantum Probab. Relat. Top., 4, No. 4, 489–509 (2001).

    Article  MathSciNet  Google Scholar 

  8. É. Yu. Shamarova and N. N. Shamarov, “Differential forms on locally convex spaces and the Stokes formula,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 84–97 (2016).

  9. O. G. Smolyanov, “De Rham flows and the Stokes formula in Hilbert spaces,” Dokl. Akad. Nauk SSSR. 286, No. 3, 554–558 (1986).

    MathSciNet  Google Scholar 

  10. Yu. V. Bogdanskii and E. V. Moravetskaya, “Surface measures on Banach manifolds with uniform structure,” Ukr. Mat. Zh., 69, No. 8, 1030–1048 (2017); English translation: Ukr. Math. J., 69, No. 8, 1196–1219 (2018).

  11. Yu. V. Bogdanskii and E. V. Moravetskaya, “Transitivity of the surface measures on Banach manifolds with uniform structure,” Ukr. Mat. Zh., 69, No. 10, 1299–1309 (2017); English translation: Ukr. Math. J., 69, No. 10, 1507–1536 (2018).

  12. S. Lang, Introduction to Differentiable Manifolds [Russian translation], Mir, Moscow (1967).

    Google Scholar 

  13. Yu. L. Daletskii and Ya. I. Belopol’skaya, Stochastic Equations and Differential Geometry [in Russian], Vyshcha Shkola, Kiev (1989).

  14. Yu. V. Bogdanskii and A. Yu. Potapenko, “Laplacian with respect to a measure on the Riemannian manifold and the Dirichlet problem. I,” Ukr. Mat. Zh., 68, No. 7, 897–907 (2016); English translation: Ukr. Math. J., 68, No. 7, 1021–1033 (2016).

  15. R. Fry and S. McManus, “Smooth bump functions and the geometry of Banach spaces. A brief survey,” Exposit. Math., 20, No. 2, 143–183 (2002).

    Article  MathSciNet  Google Scholar 

  16. D. H. Fremlin, “Measurable functions and almost continuous functions,” Manuscripta Math., 33, No. 3-4, 387–405 (1981).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu. V. Bogdanskii.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 11, pp. 1455–1468, November, 2020. Ukrainian DOI: 10.37863/umzh.v72i11.2295.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogdanskii, Y.V. Stokes Formula for Banach Manifolds. Ukr Math J 72, 1677–1694 (2021). https://doi.org/10.1007/s11253-021-01880-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-021-01880-8

Navigation