Abstract
We study the Dirichlet problem at infinity for \(\mathcal A\)-harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x 0∈M(∞) to be \(\mathcal A\)-regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x 0 in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ancona, A.: Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature. Rev. Mat. Iberoamericana 10(1), 189–220 (1994)
Anderson, M.: The Dirichlet problem at infinity for manifolds of negative curvature. J. Differential Geom. 18(4), (1983), 701–721 (1984)
Anderson, M., Schoen, R.: Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. 121, 429–461 (1985)
Bishop, R., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1–49 (1969)
Cheng, S.Y.: The Dirichlet problem at infinity for nonpositively curved manifolds. Comm. Anal. Geom. 1(1), 101–112 (1993)
Choi, H.I.: Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds. Trans. Amer. Math. Soc. 281(2), 691–716 (1984)
Eberlein, P., O’Neill, B.: Visibility manifolds. Pacific J. Math. 46, 45–109 (1973)
Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole, vol. 699 of Lecture Notes in Mathematics, Springer, Berlin (1979)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford University Press, Oxford (1993)
Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27, 715–727 (1974), eratum “A correction to: Sobolev and isoperimetric inequalities for Riemannian submanifolds”, Comm. Pure Appl. Math., 28, 765–766 (1975)
Holopainen, I.: Asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds. Proc. Amer. Math. Soc. 130(11), 3393–3400 (2002)
Holopainen, I.: Quasiregular mappings and the p-Laplace operator. Contemp. Math. 338, 219–239 (2003)
Holopainen, I., Lang, U., Vähäkangas, A.: Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces. (to appear)
Holopainen, I., Rickman, S.: Classification of Riemannian manifolds in nonlinear potential theory. Potential Anal. 2, 37–66 (1993)
Pansu, P.: Cohomologie L p des variétés à courbure négative, cas du degré 1. Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1989), 95–120 (1990) Conference on Partial Differential Equations and Geometry (Torino, 1988)
Sullivan, D.: The Dirichlet problem at infinity for a negatively curved manifold. J. Differential Geom. 18, 4 (1983), 723–732 (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vähäkangas, A. Dirichlet Problem at Infinity for \(\mathcal A\)-Harmonic Functions. Potential Anal 27, 27–44 (2007). https://doi.org/10.1007/s11118-007-9051-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-007-9051-7