Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


IMPACT FACTOR 2018: 0.536

CiteScore 2018: 0.83

SCImago Journal Rank (SJR) 2018: 1.041
Source Normalized Impact per Paper (SNIP) 2018: 0.801

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
Online
ISSN
2299-3274
See all formats and pricing
More options …

The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

Marcello Lucia
  • Corresponding author
  • Department of Mathematics, College of Staten Island-CUNY, 2800 Victory Boulevard, Staten Island, NY 10314, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Michael J. Puls
  • Corresponding author
  • Department of Mathematics, John Jay College-CUNY, 524 West 59th Street, New York, NY 10019, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-06-01 | DOI: https://doi.org/10.1515/agms-2015-0008

Abstract

Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

Keywords: Dirichlet problem at infinity; metric measure space; p-harmonic function; p-parabolic; p-Royden algebra; p-weak upper gradient; (p, p)-Sobolev inequality

MSC: Primary: 31B20; Secondary: 31C25, 54E45

References

  • [1] Anders Björn and Jana Björn. Nonlinear potential theory onmetric spaces, volume17 ofEMS Tracts inMathematics. European Mathematical Society (EMS), Zürich, 2011. Google Scholar

  • [2] Moses Glasner and Richard Katz. The Royden boundary of a Riemannian manifold. Illinois J. Math., 14:488–495, 1970. Google Scholar

  • [3] Vladimir Gol0dshtein and Marc Troyanov. Axiomatic theory of Sobolev spaces. Expo. Math., 19(4):289–336, 2001. Google Scholar

  • [4] Daniel Hansevi. The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spaces. arXiv: 1311.5955, 2013. Google Scholar

  • [5] Ilkka Holopainen, Urs Lang, and Aleksi Vähäkangas. Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces. Math. Ann., 339(1):101–134, 2007. Google Scholar

  • [6] Yong Hah Lee. Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds. Math. Ann., 318(1):181–204, 2000. Google Scholar

  • [7] Yong Hah Lee. Rough isometry and p-harmonic boundaries of complete Riemannianmanifolds. Potential Anal., 23(1):83–97, 2005. Google Scholar

  • [8] Michael J. Puls. Graphs of bounded degree and the p-harmonic boundary. Pacific J. Math., 248(2):429–452, 2010. Google Scholar

  • [9] H. L. Royden. On the ideal boundary of a Riemann surface. In Contributions to the theory of Riemann surfaces, Annals of Mathematics Studies, no. 30, pages 107–109. Princeton University Press, Princeton, N. J., 1953. Google Scholar

  • [10] L. Sario and M. Nakai. Classification theory of Riemann surfaces. Die Grundlehren der mathematischen Wissenschaften, Band 164. Springer-Verlag, New York, 1970. Google Scholar

  • [11] Nageswari Shanmugalingam. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana, 16(2):243–279, 2000. Google Scholar

  • [12] Nageswari Shanmugalingam. Some convergence results for p-harmonic functions on metric measure spaces. Proc. London Math. Soc. (3), 87(1):226–246, 2003. Google Scholar

  • [13] Stephen Willard. General topology. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. Google Scholar

  • [14] Shing Tung Yau. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math., 28:201–228, 1975. Google Scholar

About the article

Received: 2015-01-23

Accepted: 2015-05-05

Published Online: 2015-06-01


Citation Information: Analysis and Geometry in Metric Spaces, Volume 3, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2015-0008.

Export Citation

© 2015 M. Lucia and M. J. Puls. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in