Advertisement

Collectanea Mathematica

, Volume 69, Issue 3, pp 407–426 | Cite as

Wolff potential estimates for Cheeger p-harmonic functions

  • Takanobu Hara
Article
  • 82 Downloads

Abstract

In this note, we give a new proof of Wolff potential estimates for Cheeger p-superharmonic functions on metric measure spaces given by Björn et al. (J Anal Math 85:339–369, 2001). Also, we extend the estimate to Poisson type equations with signed data.

Keywords

Nonlinear elliptic equations p-Laplacian Wolff potentials Metric space Doubling measure Poincarë inequality Potential theory 

Mathematics Subject Classification

31C45 35J62 31C15 31C05 

Notes

Acknowledgements

The author wishes to thank Professor Hiroaki Aikawa for suggesting this problem. The author would like to thank the reviewers for carefully reading the manuscript and the helpful comments.

References

  1. 1.
    Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, Volume 17 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2011)CrossRefGoogle Scholar
  2. 2.
    Björn, A., Marola, N.: Moser iteration for (quasi)minimizers on metric spaces. Manuscr. Math. 121(3), 339–366 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Björn, J.: Wiener criterion for Cheeger \(p\)-harmonic functions on metric spaces. In: Aikawa, H., S\(\bar{\rm u}\)gakkai, N. (eds.) Potential Theory in Matsue, pp. 103–115. Mathematics Society of Japan, Tokyo (2006)Google Scholar
  4. 4.
    Björn, J., MacManus, P., Shanmugalingam, N.: Fat sets and pointwise boundary estimates for \(p\)-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Danielli, D., Garofalo, N., Marola, N.: Local behavior of \(p\)-harmonic Green’s functions in metric spaces. Potential Anal. 32(4), 343–362 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Duzaar, F., Mingione, G.: Gradient estimates via non-linear potentials. Am. J. Math. 133(4), 1093–1149 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Grafakos, L.: Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2008)Google Scholar
  9. 9.
    Hara, T.: Weak-type estimates and potential estimates for elliptic equations with drift terms. Potential Anal. 44(1), 189–214 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hara, T.: The Wolff potential estimate for solutions to elliptic equations with signed data. Manuscr. Math. 150(1–2), 45–58 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev Spaces on Metric Measure Spaces, Volume 27 of New Mathematical Monographs. Cambridge University Press, Cambridge (2015). An approach based on upper gradientsGoogle Scholar
  12. 12.
    Keith, S.: A differentiable structure for metric measure spaces. Adv. Math. 183(2), 271–315 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kilpeläinen, T., Malý, J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(4), 591–613 (1992)MathSciNetMATHGoogle Scholar
  15. 15.
    Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172(1), 137–161 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kinnunen, J., Martio, O.: Nonlinear potential theory on metric spaces. Illinois J. Math. 46(3), 857–883 (2002)MathSciNetMATHGoogle Scholar
  17. 17.
    Kinnunen, J., Martio, O.: Sobolev space properties of superharmonic functions on metric spaces. Results Math. 44(1–2), 114–129 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscr. Math. 105(3), 401–423 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kinnunen, J., Shanmugalingam, N.: Polar sets on metric spaces. Trans. Am. Math. Soc. 358(1), 11–37 (2006). (electronic) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Korte, R., Kuusi, T.: A note on the Wolff potential estimate for solutions to elliptic equations involving measures. Adv. Calc. Var. 3(1), 99–113 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lindqvist, P., Martio, O.: Two theorems of N. Wiener for solutions of quasilinear elliptic equations. Acta Math. 155(3–4), 153–171 (1985)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  23. 23.
    Maz’ja, V.G., Havin, V.P.: A nonlinear potential theory. Uspehi Mat. Nauk 27(6), 67–138 (1972)MathSciNetGoogle Scholar
  24. 24.
    Maz’ja, V.G.: The continuity at a boundary point of the solutions of quasi-linear elliptic equations. Vestnik Leningrad. Univ. 25(13), 42–55 (1970)MathSciNetGoogle Scholar
  25. 25.
    Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104, 1–71 (1996)MathSciNetMATHGoogle Scholar
  26. 26.
    Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16(2), 243–279 (2000)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Trudinger, N.S., Wang, X.-J.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124(2), 369–410 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations