manuscripta mathematica

, Volume 121, Issue 3, pp 339–366 | Cite as

Moser iteration for (quasi)minimizers on metric spaces

  • Anders Björn
  • Niko MarolaEmail author


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.

Mathematics Subject Classification (2000)

Primary 49N60 Secondary 35J60 Secondary 49J27 


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  1. 1.
    Adams D., Hedberg L.I. (1996) Function Spaces and Potential Theory. Springer, Berlin Heidelberg New YorkGoogle Scholar
  2. 2.
    Björn, A.: A weak Kellogg property for quasiminimizers. Comment. Math. Helv. (to appear)Google Scholar
  3. 3.
    Björn A. (2005) Characterizations of p-superharmonic functions on metric spaces. Studia Math. 169, 45–62MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Björn A. (2006) Removable singularities for bounded p-harmonic and quasi(super)harmonic functions on metric spaces. Ann. Acad. Sci. Fenn. Math. 31, 71–95MathSciNetzbMATHGoogle Scholar
  5. 5.
    Björn, A., Björn, J.: Boundary regularity for p-harmonic functions and solutions of the obstacle problem. Linköping (2004)Google Scholar
  6. 6.
    Björn A., Björn J., Shanmugalingam N. (2003) The Perron method for p-harmonic functions. J. Differential Equations 195, 398–429MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. Linköping (2006)Google Scholar
  8. 8.
    Björn J. (2002) Boundary continuity for quasiminimizers on metric spaces. Illinois J. Math. 46, 383–403MathSciNetzbMATHGoogle Scholar
  9. 9.
    Björn, J.: Approximation by regular sets in metric spaces. Linköping (2004)Google Scholar
  10. 10.
    Buckley S.M. (1999) Inequalities of John–Nirenberg type in doubling spaces. J. Anal. Math. 79, 215–240MathSciNetzbMATHGoogle Scholar
  11. 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, 25–43MathSciNetGoogle Scholar
  12. 12.
    Federer H. (1969) Geometric Measure Theory. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  13. 13.
    Giaquinta M. (1983) Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton Univ. Press, PrincetonzbMATHGoogle Scholar
  14. 14.
    Giaquinta M., Giusti E. (1982) On the regularity of the minima of variational integrals. Acta Math. 148, 31–46MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Giaquinta M., Giusti E. (1984) Quasi-minima. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 79–107MathSciNetzbMATHGoogle Scholar
  16. 16.
    Giusti E. (2003) Direct Methods in the Calculus of Variations. World Scientific, SingaporezbMATHGoogle Scholar
  17. 17.
    Hajłasz P., Koskela P. (1995) Sobolev meets Poincaré. C. R. Acad. Sci. Paris 320, 1211–1215zbMATHGoogle Scholar
  18. 18.
    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000)Google Scholar
  19. 19.
    Hedberg L.I. (1981) Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. Acta Math. 147, 237–264MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hedberg, L.I., Netrusov, Yu.: An axiomatic approach to function spaces, spectral synthesis and Luzin approximation. Mem. Amer. Math. Soc. (to appear)Google Scholar
  21. 21.
    Heinonen J. (2001) Lectures on Analysis on Metric Spaces. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  22. 22.
    Heinonen J., Kilpeläinen T., Martio O. (1993) Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, OxfordzbMATHGoogle Scholar
  23. 23.
    Keith S. (2003) Modulus and the Poincaré inequality on metric measure spaces. Math. Z. 245, 255–292MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Preprint, 2005Google Scholar
  25. 25.
    Kilpeläinen T., Kinnunen J., Martio O. (2000) Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233–247MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kinnunen J., Martio O. (1996) The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21, 367–382MathSciNetzbMATHGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Kinnunen J., Martio O. (2002) Nonlinear potential theory on metric spaces. Illinois Math. J. 46, 857–883MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kinnunen J., Martio O. (2003) Potential theory of quasiminimizers. Ann. Acad. Sci. Fenn. Math. 28, 459–490MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kinnunen J., Martio O. (2003) Sobolev space properties of superharmonic functions on metric spaces. Results Math. 44, 114–129MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kinnunen J., Shanmugalingam N. (2001) Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kinnunen, J., Shanmugalingam, N.: Private communication (2005)Google Scholar
  33. 33.
    Koskela P. (1999) Removable sets for Sobolev spaces. Ark. Mat. 37, 291–304MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Koskela P., MacManus P. (1998) Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17MathSciNetzbMATHGoogle Scholar
  35. 35.
    Marola, N.: Moser’s method for minimizers on metric measure spaces. Preprint A478, Helsinki University of Technology, Institute of Mathematics (2004)Google Scholar
  36. 36.
    Mateu J., Mattila P., Nicolau A., Orobitg J. (2000) BMO for nondoubling measures. Duke Math. J. 102, 533–565MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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, 457–468MathSciNetzbMATHGoogle Scholar
  38. 38.
    Moser J. (1961) On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rudin W. (1987) Real and Complex Analysis, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  40. 40.
    Shanmugalingam N. (2000) Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279MathSciNetzbMATHGoogle Scholar
  41. 41.
    Shanmugalingam N. (2001). Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050MathSciNetzbMATHGoogle Scholar
  42. 42.
    Tolksdorf P. (1986) Remarks on quasi(sub)minima. Nonlinear Anal. 10, 115–120MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ziemer W.P. (1986) Boundary regularity for quasiminima. Arch. Rational Mech. Anal. 92, 371–382MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsLinköpings UniversitetLinköpingSweden
  2. 2.Institute of MathematicsHelsinki University of TechnologyHelsinkiFinland

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