ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

Title & Authors
ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS
Kim, Seok-Woo; Lee, Yong-Hah;

Abstract
We prove that for any continuous function f on the s-harmonic $\small{(1}$<$\small{s}$<$\small{{\infty})}$ boundary of a complete Riemannian manifold M, there exists a solution, which is a limit of a sequence of bounded energy finite solutions in the sense of supremum norm, for a certain elliptic operator A on M whose boundary value at each s-harmonic boundary point coincides with that of f. If $\small{E_1,\;E_2,...,E_{\iota}}$ are s-nonparabolic ends of M, then we also prove that there is a one to one correspondence between the set of bounded energy finite solutions for A on M and the Cartesian product of the sets of bounded energy finite solutions for A on $\small{E_i}$ which vanish at the boundary $\small{{\partial}E_{\iota}\;for\;{\iota}=1,2,...,{\iota}}$
Keywords
s-harmonic boundary;A-harmonic function;end;
Language
English
Cited by
1.
The connectivity at infinity of a manifold and Lq,p-Sobolev inequalities, Expositiones Mathematicae, 2014, 32, 4, 365
References
1.
H. Donnelly, Bounded harmonic functions and positive Ricci curvature, Math. Z. 191 (1986), no. 4, 559-565

2.
A. A. Grigor'yan, On the set of positive solutions of the Laplace-Beltrami equation on Riemannian manifolds of a special form, Izv. Vyssh. Uchebn. Zaved., Matematika (1987) no.2, 30-37: English transl. Soviet Math. (Iz, VUZ) 31 (1987) no.2, 48-60

3.
J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993

4.
E. Hewitt and K. Stormberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer-Verlag, New York 1965

5.
I. Holopainen, Solutions of elliptic equations on manifolds with roughly Euclidean ends, Math. Z. 217 (1994), no. 3, 459-477

6.
S. W. Kim and Y. H. Lee, Rough isometry, harmonic functions and harmonic maps on a complete Riemannian manifold, J. Korean Math. Soc. 36 (1999), no. 1, 73-95

7.
S. W. Kim and Y. H. Lee, Generalized Liouville property for Schrodinger operator on Riemannian manifolds, Math. Z. 238 (2001), no. 2, 355-387

8.
Y. H. Lee, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann. 318 (2000), no. 1, 181-204

9.
P. Li and L-F. Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), no. 1, 171-207

10.
P. Li and L-F. Tam, Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), no. 2, 277-318

11.
J. Maly and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, 51, American Mathematical Society, Providence, RI, 1997

12.
C. J. Sung, L. F. Tam, and J. Wang, Spaces of harmonic functions, J. London Math. Soc. (2) 61 (2000), no. 3, 789-806

13.
S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228