Skip to main content
SpringerLink
Log in

L 3/2-estimates of vector fields with L 1 curl in a bounded domain

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

This paper establishes the estimates of L 3/2 norm of the vector fields in a bounded domain with vanishing tangential component on the boundary, in terms of the L 1 norm of the curl, the negative exponent Sobolev norm of the divergence, and on some quantities depending on the topology of the domain. As the similar proof we also obtain the estimates of L p norm of the vector fields in terms of the negative exponent Sobolev norms of the curl and divergence.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Amrouche C., Girault V., Giroire J.: Weighted Sobolev spaces for the Laplace equation in \({\mathbb{R}^n}\) . J. Pure Appl. Math. 73(6), 579–606 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Amrouche C., Nguyen H.H.: New estimates for the div-curl-grad operators and elliptic problems with L 1s-data in the whole space and in the half-space. J. Differ. Equ. 250(6), 3150–3195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgain J., Brezis H.: New estimates for the Laplacian, the div-curl, and related Hodge systems. C. R. Math. Acad. Sci. Paris 338, 539–543 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bates, P., Pan, X.B.: Nucleation of instability in Meissner state of 3-dimensional superconductors. Commun. Math. Phys. 276, 571–610 (2007). Erratum, 283, 861 (2008)

    Google Scholar 

  6. Bolik J., Wahl W. von: Estimating \({\nabla\bf{u}}\) in terms of div u, curl u and either (ν, u) or ν × u and the topology. Math. Methods Appl. Sci. 20, 737–744 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calderón A.P., Zygmund A.: On singular integrals. Am. J. Math. 78(2), 289–309 (1956)

    Article  MATH  Google Scholar 

  8. Dautray R., Lions J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, New York (1990)

    Book  Google Scholar 

  9. Heinemann, U.: Die regularisierende Wirkung der Randintegraloperatoren der klassischen Potentialtheorie in den Räumen hölderstetiger Funktionen. Diplomarbeit, Bayreuth (1992)

  10. Kardestuncer, H., Norrie, D.H. (eds.): Finite Element Handbook. McGraw-Hill. New York, Toronto (1987)

    Google Scholar 

  11. Kozono H., Yanagisawa T.: L r-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains. Indiana Univ. Math. J. 58(4), 1853–1920 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Monneau R.: Quasilinear elliptic systems arising in a three-dimensional type-II superconductor for infinite κ. Nonlinear Anal. 52, 917–930 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Neudert M., Wahl W. von: Asymptotic behaviour of the div–curl problem in exterior domains. Adv. Differ. Equ. 6, 1347–1376 (2001)

    MATH  Google Scholar 

  14. Wahl, W. von: Vorlesung über das Außenraumproblem für die instationären Gleichungen von Navier–Stokes. Rudolph-Lipschitz-Vorlesung, Sonderforschungsbereich 256 Nichtlineare partielle Differentialgleichungen, Bonn (1989)

  15. Wahl W. von: Abschätzungen für das Neumann-Problem und die Helmholtz-Zerlegung von L p. Nachr. Ak. d. Wiss. Göttingen. II. Mathematisch-Physikalische Klasse 2, 1–29 (1990)

    Google Scholar 

  16. Wahl W. von: Estimating \({\nabla{\bf u}}\) by div u and curl u. Math. Methods Appl. Sci. 15, 123–143 (1990)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, People’s Republic of China

    Xingfei Xiang

Authors
  1. Xingfei Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xingfei Xiang.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiang, X. L 3/2-estimates of vector fields with L 1 curl in a bounded domain. Calc. Var. 46, 55–74 (2013). https://doi.org/10.1007/s00526-011-0473-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-011-0473-0

Mathematics Subject Classification (2000)

Search

Navigation