Abstract and Applied Analysis

Strongly nonlinear potential theory on metric spaces

Noureddine Aïssaoui

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Abstract

We define Orlicz-Sobolev spaces on an arbitrary metric space with a Borel regular outer measure, and we develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. We prove that each Orlicz-Sobolev function has a quasi-continuous representative. We give estimates for the capacity of balls when the measure is doubling. Under additional regularity assumption on the measure, we establish some relations between capacity and Hausdorff measures.

Article information

Source
Abstr. Appl. Anal., Volume 7, Number 7 (2002), 357-374.

Dates
First available in Project Euclid: 14 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1050348397

Digital Object Identifier
doi:10.1155/S1085337502203024

Mathematical Reviews number (MathSciNet)
MR1939129

Zentralblatt MATH identifier
1017.46022

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 31B15: Potentials and capacities, extremal length
Secondary: 28A80: Fractals [See also 37Fxx]

Citation

Aïssaoui, Noureddine. Strongly nonlinear potential theory on metric spaces. Abstr. Appl. Anal. 7 (2002), no. 7, 357--374. doi:10.1155/S1085337502203024. https://projecteuclid.org/euclid.aaa/1050348397


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References

  • R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  • N. Aïssaoui, Note sur la capacitabilité dans les espaces d'Orlicz, Ann. Sci. Math. Québec 19 (1995), no. 2, 107–113 (French).
  • ––––, Bessel potentials in Orlicz spaces, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 1, 55–79.
  • ––––, Instability of capacity in Orlicz spaces, Potential Anal. 6 (1997), no. 4, 327–346.
  • ––––, Some developments of strongly nonlinear potential theory, Libertas Math. 19 (1999), 155–170.
  • N. Aïssaoui and A. Benkirane, Capacités dans les espaces d'Orlicz, Ann. Sci. Math. Québec 18 (1994), no. 1, 1–23 (French).
  • ––––, Potentiel non linéaire dans les espaces d'Orlicz, Ann. Sci. Math. Québec 18 (1994), no. 2, 105–118 (French).
  • H. Federrer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.
  • B. Franchi, P. Hajłasz, and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1903–1924.
  • D. Gallardo, Orlicz spaces for which the Hardy-Littlewood maximal operator is bounded, Publ. Mat. 32 (1988), no. 2, 261–266.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
  • P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no. 4, 403–415.
  • P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101.
  • P. Hajłasz and O. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, J. Funct. Anal. 143 (1997), no. 1, 221–246.
  • L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.
  • T. Kilpeläinen, J. Kinnunen, and O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 (2000), no. 3, 233–247.
  • J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367–382.
  • M. A. Krasnosel'skiĭ and Ja. B. Rutickiĭ, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, 1961.
  • A. Kufner, O. John, and S. Fučík, Function Spaces, Noordhoff International Publishing, Leyden, 1977.
  • W. A. J. Luxemburg, Banach Function Spaces, Thesis, Technische Hogeschool te Delft, 1955.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
  • L. Simon, Lectures on Geometric Measure Theory, Australian National University Centre for Mathematical Analysis, Canberra, 1983.
  • W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989.