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On Null Sets of P-Harmonic Measures

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Partial Differential Equations with Minimal Smoothness and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 42))

Abstract

In a Lipschitz domain Ω ⊂ R n, associated to the p-Laplace equation

$$\text{div}(\left\vert\bigtriangledown_u\right\vert^{p-2}\bigtriangledown_u)=0$$

one can define a notion of p-harmonic measure on subsets E ⊂ ∂Ω by solving the Dirichlet problem for (1) with boundary values χE. Denote by w p(x; E) the p-harmbnic function with boundary values 1 on E and 0 on ∂Ω\E. In the linear case p = 2 for each x ∈ Ω we do obtain a Borei measure on ∂Ω. This is no longer true in the nonlinear case p ≠ 2. Yet the monotonicity properties of classical harmonic measures extend to their nonlinear counterparts. Many applications of this principle are in [GLM], [HM].

Research partially supported by NSF Grants DMS-8901695 and DMS-8901524.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, Illinois, 61801, USA

    Patricio Avilés

  2. Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Juan J. Manfredi

Authors
  1. Patricio Avilés
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  2. Juan J. Manfredi
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29303, USA

    B. Dahlberg

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    R. Fefferman , Carlos Kenig  & J. Pipher ,  & 

  3. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Eugene Fabes

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    David Jerison

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© 1992 Springer-Verlag New York, Inc.

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Avilés, P., Manfredi, J.J. (1992). On Null Sets of P-Harmonic Measures. In: Dahlberg, B., Fefferman, R., Kenig, C., Fabes, E., Jerison, D., Pipher, J. (eds) Partial Differential Equations with Minimal Smoothness and Applications. The IMA Volumes in Mathematics and its Applications, vol 42. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2898-1_4

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  • DOI: https://doi.org/10.1007/978-1-4612-2898-1_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7712-5

  • Online ISBN: 978-1-4612-2898-1

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