Revista Matemática Iberoamericana

Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces

Anders Björn , Jana Björn , and Mikko Parviainen

Full-text: Open access

Abstract

We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have $L^q$-Lebesgue points everywhere.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 1 (2010), 147-174.

Dates
First available in Project Euclid: 16 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1266330121

Mathematical Reviews number (MathSciNet)
MR2666312

Zentralblatt MATH identifier
1203.31018

Subjects
Primary: 31C45: Other generalizations (nonlinear potential theory, etc.)
Secondary: 31C05: Harmonic, subharmonic, superharmonic functions 35J60: Nonlinear elliptic equations

Keywords
$mathcal{A}$-harmonic fundamental convergence theorem Lebesgue point metric space Newtonian function nonlinear $p$-harmonic quasicontinuous Sobolev function superharmonic superminimizer supersolution weak upper gradient

Citation

Björn, Anders; Björn, Jana; Parviainen, Mikko. Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces. Rev. Mat. Iberoamericana 26 (2010), no. 1, 147--174. https://projecteuclid.org/euclid.rmi/1266330121


Export citation

References

  • Björn, A.: Characterizations of \p-superharmonic functions on metric spaces. Studia Math. 169 (2005), no. 1, 45-62.
  • Björn, A.: A weak Kellogg property for quasiminimizers. Comment. Math. Helv. 81 (2006), no. 4, 809-825.
  • Björn, A.: A regularity classification of boundary points for $p$-harmonic functions and quasiminimizers. J. Math. Anal. Appl. 338 (2008), no. 1, 39-47.
  • Björn, A. and Björn, J.: Boundary regularity for $p$-harmonic functions and solutions of the obstacle problem. J. Math. Soc. Japan 58 (2006), no. 4, 1211-1232.
  • Björn, A., Björn, J., Mäkäläinen, T. and Parviainen, M.: Nonlinear balayage on metric spaces. Nonlinear Anal. 71 (2009), no. 5-6, 2153-2171.
  • Björn, A., Björn, J. and Shanmugalingam, N.: The Dirichlet problem for $p$-harmonic functions on metric spaces. J. Reine Angew. Math. 556 (2003), 173-203.
  • Björn, A., Björn, J. and Shanmugalingam, N.: The Perron method for $p$-harmonic functions. J. Differential Equations 195 (2003), no. 2, 398-429.
  • Björn, A., Björn, J. and Shanmugalingam, N.: Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces. Houston J. Math. 34 (2008), no. 4, 1197-1211.
  • Björn, A. and Marola, N.: Moser iteration for (quasi)minimizers on metric spaces. Manuscripta Math. 121 (2006), no. 3, 339-366.
  • Björn, J.: Boundary continuity for quasiminimizers on metric spaces. Illinois J. Math. 46 (2002), no. 2, 383-403.
  • Björn, J.: Fine continuity on metric spaces. Manuscripta Math. 125 (2008), no. 3, 369-381.
  • Björn, J., MacManus, P. and Shanmugalingam, N.: Fat sets and pointwise boundary estimates for $p$-harmonic functions in metric spaces. J. Anal. Math. 85 (2001), 339-369.
  • Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), no. 3, 428-517.
  • Crandall, M.G. and Zhang, J. Another way to say harmonic. Trans. Amer. Math. Soc. 355 (2003), no. 1, 241-263.
  • Doob, J.L.: Classical potential theory and its probabilistic counterpart. Grundlehren der Mathematischen Wissenschaften 262. Springer-Verlag, New York, 1984.
  • Evans, L.C. and Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Math. CRC Press, Boca Raton, FL, 1992.
  • Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ, 1983.
  • Hajłasz, P. and Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.
  • Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
  • Heinonen, J.: Nonsmooth calculus. Bull. Amer. Math. Soc. 44 (2007), no. 2, 163-232.
  • Heinonen, J. and Kilpeläinen, T.: Polar sets for supersolutions of degenerate elliptic equations. Math. Scand. 63 (1988), no. 1, 136-150.
  • Heinonen, J. and Kilpeläinen, T.: On the Wiener criterion and quasilinear obstacle problems. Trans. Amer. Math. Soc. 310 (1988), no. 1, 239-255.
  • Heinonen, J., Kilpeläinen, T. and Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Dover Pubs., Mineola, NY, 2006.
  • Heinonen, J. and Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), no. 1, 1-61.
  • Kallunki, S. and Shanmugalingam, N.: Modulus and continuous capacity. Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 455-464.
  • Keith, S. and Zhong, X.: The Poincaré inequality is an open ended condition. Ann. of Math. (2) 167 (2008), no. 2, 575-599.
  • Kilpeläinen, T.: Potential theory for supersolutions of degenerate elliptic equations. Indiana Univ. Math. J. 38 (1989), no. 2, 253-275.
  • Kilpeläinen, T.: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 95-113.
  • Kilpeläinen, T., Kinnunen, J. and Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), no. 3, 233-247.
  • Kinnunen, J. and Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 18 (2002), no. 3, 685-700.
  • Kinnunen, J. and Martio, O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367-382.
  • Kinnunen, J. and Martio, O.: Choquet property for the Sobolev capacity in metric spaces. In Proceedings on Analysis and Geometry (Novosibirsk, Akademgorodok, 1999), 285-290. Sobolev Institute Press, Novosibirsk, 2000.
  • Kinnunen, J. and Martio, O.: Nonlinear potential theory on metric spaces. Illinois J. Math. 46 (2002), no. 3, 857-883.
  • Kinnunen, J. and Martio, O.: Potential theory of quasiminimizers. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 459-490.
  • Kinnunen, J. and Martio, O.: Sobolev space properties of superharmonic functions on metric spaces. Results Math. 44 (2003), no. 1-2, 114-129.
  • Kinnunen, J. and Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105 (2001), no. 3, 401-423.
  • Kinnunen, J. and Shanmugalingam, N.: Polar sets on metric spaces. Trans. Amer. Math. Soc. 358 (2006), no. 1, 11-37.
  • Koskela, P. and MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131 (1998), no. 1, 1-17.
  • Kuusi, T.: Lower semicontinuity of weak supersolutions to a nonlinear parabolic equation. Differential Integral Equations 22 (2009), no. 11-12, 1211-1222.
  • Malý, J. and Ziemer, W.P.: Fine regularity of solutions of elliptic partial differential equations. Mathematical Surveys and Monographs 51. Amererican Mathematical Society, Providence, RI, 1997.
  • Ono, T.: Private communication, 2004.
  • Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16 (2000), 243-279.
  • Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45 (2001), no. 3, 1021-1050.
  • Shanmugalingam, N.: Some convergence results for $p$-harmonic functions on metric measure spaces. Proc. London Math. Soc. (3) 87 (2003), no. 1, 226-246.
  • Yosida, K.: Functional analysis. Grundlehren der Mathematischen Wissenschaften 123. Springer-Verlag, Berlin-New York, 1980.