Rough isometry and p-harmonic boundaries of complete Riemannian manifolds

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Abstract

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds.

Original languageEnglish
Pages (from-to)83-97
Number of pages15
JournalPotential Analysis
Volume23
Issue number1
DOIs
StatePublished - Aug 2005

Keywords

  • p-harmonic boundary
  • p-harmonic function
  • Rough isometry

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