Dirichlet finite harmonic functions and points at infinity of graphs and manifolds

Dirichlet finite harmonic functions and points at infinity of graphs and manifolds

Tae Hattori and Atsushi Kasue

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 83, Number 7 (2007), 129-134.

Abstract

In this paper, we consider the Royden compactifications relative to $p$-Dirichlet integrals of infinite graphs and noncompact Riemannian manifolds, and study the behavior of rough isometries in the compactifications, proving bijective correspondence of the spaces of $p$-harmonic functions with finite $p$-energy.

First Page: Show

Primary Subjects: 53C21, 58D17, 58J50

Keywords: graph; Riemannian manifold; $p$-energy; rough isometry; Royden’s compactification

Full-text: Open access

PDF File (119 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1200672014
Mathematical Reviews number (MathSciNet): MR2361425
Digital Object Identifier: doi:10.3792/pjaa.83.129
Zentralblatt MATH identifier: 1145.53310

back to Table of Contents

References

T. Coulhon and L. Saloff-Coste, Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana 11 (1995), no. 3, 687--726.

Mathematical Reviews (MathSciNet): MR1363211

T. Coulhon, I. Holopainen and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic $p$-Laplacians with applications to Picard type theorems, Geom. Funct. Anal. 11 (2001), no. 6, 1139--1191.

Mathematical Reviews (MathSciNet): MR1878317

Digital Object Identifier: doi:10.1007/s00039-001-8227-3

J. Eells and B. Fuglede, Harmonic maps between Riemannian polyhedra, Cambridge Univ. Press, Cambridge, 2001.

Mathematical Reviews (MathSciNet): MR1848068

Zentralblatt MATH: 0979.31001

P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.

I. Holopainen, Rough isometries and $p$-harmonic functions with finite Dirichlet integral, Rev. Mat. Iberoamericana 10 (1994), no. 1, 143--176.

Mathematical Reviews (MathSciNet): MR1271760

I. Holopainen and P. M. Soardi, $p$-harmonic functions on graphs and manifolds, Manuscripta Math. 94 (1997), no. 1, 95--110.

Mathematical Reviews (MathSciNet): MR1468937

M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391--413.

Mathematical Reviews (MathSciNet): MR792983

M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), no. 2, 227--238.

Mathematical Reviews (MathSciNet): MR833199

M. Kanai, Analytic inequalities, and rough isometries between noncompact Riemannian manifolds, in Curvature and topology of Riemannian manifolds (Katata, 1985), 122--137, Lecture Notes in Math., 1201, Springer, Berlin.

Mathematical Reviews (MathSciNet): MR859579

Digital Object Identifier: doi:10.1007/BFb0075650

A. Kasue, Convergence of metric graphs and energy forms. (Preprint).

K. Kuwae, Y. Machigashira and T. Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269--316.

Mathematical Reviews (MathSciNet): MR1865418

Digital Object Identifier: doi:10.1007/s002090100252

Zentralblatt MATH: 1001.53017

Y. H. Lee, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann. 318 (2000), no. 1, 181--204.

Mathematical Reviews (MathSciNet): MR1785581

Digital Object Identifier: doi:10.1007/s002080000118

Zentralblatt MATH: 0968.58018

Y. H. Lee, Rough isometry and $p$-harmonic boundaries of complete Riemannian manifolds, Potential Anal. 23 (2005), no. 1, 83--97.

Mathematical Reviews (MathSciNet): MR2136210

Digital Object Identifier: doi:10.1007/s11118-004-3261-z

Zentralblatt MATH: 1082.31005

M. Nakai, Potential theory on Royden compactifications, Bull. Nagoya Inst. Tech. 47 (1995), 171--191. (in Japanese).

Mathematical Reviews (MathSciNet): MR1387935

M. Nakai, Existence of quasi-isometric mappings and Royden compactifications, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 1, 239--260.

Mathematical Reviews (MathSciNet): MR1737436

H. Tanaka, Harmonic boundaries of Riemannian manifolds, Nonlinear Anal. 14 (1990), no. 1, 55--67.

Mathematical Reviews (MathSciNet): MR1028247

L. Sario and M. Nakai, Classification theory of Riemann surfaces, Springer, New York, 1970.

Mathematical Reviews (MathSciNet): MR264064

Zentralblatt MATH: 0199.40603

P. M. Soardi, Potential theory on infinite networks, Lecture Notes in Math., 1590, Springer, Berlin, 1994.

Mathematical Reviews (MathSciNet): MR1324344

Zentralblatt MATH: 0818.31001

previous :: next