Skip to main content

Advertisement

SpringerLink
Log in

Moser iteration for (quasi)minimizers on metric spaces

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.

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 D., Hedberg L.I. (1996) Function Spaces and Potential Theory. Springer, Berlin Heidelberg New York

    Google Scholar 

  2. Björn, A.: A weak Kellogg property for quasiminimizers. Comment. Math. Helv. (to appear)

  3. Björn A. (2005) Characterizations of p-superharmonic functions on metric spaces. Studia Math. 169, 45–62

    Article  MathSciNet  MATH  Google Scholar 

  4. Björn A. (2006) Removable singularities for bounded p-harmonic and quasi(super)harmonic functions on metric spaces. Ann. Acad. Sci. Fenn. Math. 31, 71–95

    MathSciNet  MATH  Google Scholar 

  5. Björn, A., Björn, J.: Boundary regularity for p-harmonic functions and solutions of the obstacle problem. Linköping (2004)

  6. Björn A., Björn J., Shanmugalingam N. (2003) The Perron method for p-harmonic functions. J. Differential Equations 195, 398–429

    Article  MathSciNet  MATH  Google Scholar 

  7. Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. Linköping (2006)

  8. Björn J. (2002) Boundary continuity for quasiminimizers on metric spaces. Illinois J. Math. 46, 383–403

    MathSciNet  MATH  Google Scholar 

  9. Björn, J.: Approximation by regular sets in metric spaces. Linköping (2004)

  10. Buckley S.M. (1999) Inequalities of John–Nirenberg type in doubling spaces. J. Anal. Math. 79, 215–240

    MathSciNet  MATH  Google Scholar 

  11. De Giorgi E. (1957) Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. 3, 25–43

    MathSciNet  Google Scholar 

  12. Federer H. (1969) Geometric Measure Theory. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  13. Giaquinta M. (1983) Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton Univ. Press, Princeton

    MATH  Google Scholar 

  14. Giaquinta M., Giusti E. (1982) On the regularity of the minima of variational integrals. Acta Math. 148, 31–46

    Article  MathSciNet  MATH  Google Scholar 

  15. Giaquinta M., Giusti E. (1984) Quasi-minima. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 79–107

    MathSciNet  MATH  Google Scholar 

  16. Giusti E. (2003) Direct Methods in the Calculus of Variations. World Scientific, Singapore

    MATH  Google Scholar 

  17. Hajłasz P., Koskela P. (1995) Sobolev meets Poincaré. C. R. Acad. Sci. Paris 320, 1211–1215

    MATH  Google Scholar 

  18. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000)

  19. Hedberg L.I. (1981) Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. Acta Math. 147, 237–264

    Article  MathSciNet  MATH  Google Scholar 

  20. Hedberg, L.I., Netrusov, Yu.: An axiomatic approach to function spaces, spectral synthesis and Luzin approximation. Mem. Amer. Math. Soc. (to appear)

  21. Heinonen J. (2001) Lectures on Analysis on Metric Spaces. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  22. Heinonen J., Kilpeläinen T., Martio O. (1993) Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford

    MATH  Google Scholar 

  23. Keith S. (2003) Modulus and the Poincaré inequality on metric measure spaces. Math. Z. 245, 255–292

    Article  MathSciNet  MATH  Google Scholar 

  24. Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Preprint, 2005

  25. Kilpeläinen T., Kinnunen J., Martio O. (2000) Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233–247

    Article  MathSciNet  Google Scholar 

  26. Kinnunen J., Martio O. (1996) The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21, 367–382

    MathSciNet  MATH  Google Scholar 

  27. Kinnunen, J., Martio, O.: Choquet property for the Sobolev capacity in metric spaces. In: Proceedings on Analysis and Geometry (Novosibirsk, Akademgorodok, 1999), pp. 285–290, Sobolev Institute Press, Novosibirsk (2000)

  28. Kinnunen J., Martio O. (2002) Nonlinear potential theory on metric spaces. Illinois Math. J. 46, 857–883

    MathSciNet  MATH  Google Scholar 

  29. Kinnunen J., Martio O. (2003) Potential theory of quasiminimizers. Ann. Acad. Sci. Fenn. Math. 28, 459–490

    MathSciNet  MATH  Google Scholar 

  30. Kinnunen J., Martio O. (2003) Sobolev space properties of superharmonic functions on metric spaces. Results Math. 44, 114–129

    MathSciNet  MATH  Google Scholar 

  31. Kinnunen J., Shanmugalingam N. (2001) Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423

    Article  MathSciNet  MATH  Google Scholar 

  32. Kinnunen, J., Shanmugalingam, N.: Private communication (2005)

  33. Koskela P. (1999) Removable sets for Sobolev spaces. Ark. Mat. 37, 291–304

    Article  MathSciNet  MATH  Google Scholar 

  34. Koskela P., MacManus P. (1998) Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17

    MathSciNet  MATH  Google Scholar 

  35. Marola, N.: Moser’s method for minimizers on metric measure spaces. Preprint A478, Helsinki University of Technology, Institute of Mathematics (2004)

  36. Mateu J., Mattila P., Nicolau A., Orobitg J. (2000) BMO for nondoubling measures. Duke Math. J. 102, 533–565

    Article  MathSciNet  MATH  Google Scholar 

  37. Moser J. (1960) A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13, 457–468

    MathSciNet  MATH  Google Scholar 

  38. Moser J. (1961) On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591

    MathSciNet  MATH  Google Scholar 

  39. Rudin W. (1987) Real and Complex Analysis, 3rd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  40. Shanmugalingam N. (2000) Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279

    MathSciNet  MATH  Google Scholar 

  41. Shanmugalingam N. (2001). Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050

    MathSciNet  MATH  Google Scholar 

  42. Tolksdorf P. (1986) Remarks on quasi(sub)minima. Nonlinear Anal. 10, 115–120

    Article  MathSciNet  Google Scholar 

  43. Ziemer W.P. (1986) Boundary regularity for quasiminima. Arch. Rational Mech. Anal. 92, 371–382

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Linköpings Universitet, 581 83, Linköping, Sweden

    Anders Björn

  2. Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FI-02015 TKK, Helsinki, Finland

    Niko Marola

Authors
  1. Anders Björn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Niko Marola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Niko Marola.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Björn, A., Marola, N. Moser iteration for (quasi)minimizers on metric spaces. manuscripta math. 121, 339–366 (2006). https://doi.org/10.1007/s00229-006-0040-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0040-8

Mathematics Subject Classification (2000)

Search

Navigation