Skip to main content

Advertisement

SpringerLink
Log in

Quasiconformal extension fields

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

Abstract

We consider extensions of differential fields of mappings and obtain a lower bound for energy of quasiconformal extension fields in terms of the topological degree. We also consider the related minimization problem for the q-harmonic energy, and show that the energy minimizers admit higher integrability.

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. Heinonen, J., Keith, S.: Flat forms, bi-lipschitz parametrizations, and smoothability of manifolds. ArXiv e-prints (Sept 2009)

  2. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1993)

  3. Heinonen J., Pankka P., Rajala K.: Quasiconformal frames. Arch. Ration. Mech. Anal. 196(3), 839–866 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Heinonen J., Sullivan D.: On the locally branched Euclidean metric gauge. Duke Math. J. 114(1), 15–41 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Iwaniec T.,  Lutoborski A.: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125(1), 25–79 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Iwaniec T., Martin G.: Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2001)

    Google Scholar 

  7. Kirby R.C.: Stable homeomorphisms and the annulus conjecture. Ann. Math. (2) 89, 575–582 (1969)

    Article  MathSciNet  Google Scholar 

  8. Kirby, R.C., Siebenmann, L.C.: Foundational essays on topological manifolds, smoothings, and triangulations. With Notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, NJ (1977)

  9. Martio O., Ziemer W.P.: Lusin’s condition (N) and mappings with nonnegative Jacobians. Mich. Math. J. 39(3), 495–508 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Reshetnyak Y.G.: Some geometric properties of functions and mappings with generalized derivatives. Sibirsk. Mat. Zh. 7, 886–919 (1966)

    Google Scholar 

  11. Reshetnyak, Y.G.: Space mappings with bounded distortion. Translations of Mathematical Monographs, vol. 73. Translated from the Russian by H. H. McFaden. American Mathematical Society, Providence, RI, (1989)

  12. Rickman, S.: Quasiregular mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26. Springer, Berlin (1993)

  13. Sullivan, D.: Hyperbolic geometry and homeomorphisms. In: Geometric Topology. Proceedings of Georgia Topology Conference, Athens, GA, 1977, pp. 543–555. Academic Press, New York (1979)

  14. Sullivan, D.: Exterior d, the local degree, and smoothability. In: Prospects in Topology, Princeton, NJ, 1994. Annals of Mathematics Studies, vol. 138, pp. 328–338. Princeton University Press, Princeton, NJ (1995)

  15. Tukia P., Väisälä J.: Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(2), 303–342 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, Helsinki, 00014, Finland

    Pekka Pankka

  2. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, Jyväskylä, 40014, Finland

    Kai Rajala

Authors
  1. Pekka Pankka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Rajala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pekka Pankka.

Additional information

Communicated by J. Ball.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pankka, P., Rajala, K. Quasiconformal extension fields. Calc. Var. 42, 73–91 (2011). https://doi.org/10.1007/s00526-010-0380-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-010-0380-9

Mathematics Subject Classification (2000)

Search

Navigation