# ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

• Kim, Seok-Woo ;
• Lee, Yong-Hah
• Published : 2008.05.31
• 62 3

#### Abstract

We prove that for any continuous function f on the s-harmonic $(1<s<{\infty})$ 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 $E_1,\;E_2,...,E_{\iota}$ 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 $E_i$ which vanish at the boundary ${\partial}E_{\iota}\;for\;{\iota}=1,2,...,{\iota}$

#### Keywords

s-harmonic boundary;A-harmonic function;end

#### 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. 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
4. I. Holopainen, Solutions of elliptic equations on manifolds with roughly Euclidean ends, Math. Z. 217 (1994), no. 3, 459-477 https://doi.org/10.1007/BF02571955
5. 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
6. S. W. Kim and Y. H. Lee, Generalized Liouville property for Schrodinger operator on Riemannian manifolds, Math. Z. 238 (2001), no. 2, 355-387 https://doi.org/10.1007/s002090100257
7. Y. H. Lee, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann. 318 (2000), no. 1, 181-204 https://doi.org/10.1007/s002080000118
8. 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 https://doi.org/10.2307/1971292
9. P. Li and L-F. Tam, Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), no. 2, 277-318 https://doi.org/10.4310/jdg/1214456219
10. 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
11. C. J. Sung, L. F. Tam, and J. Wang, Spaces of harmonic functions, J. London Math. Soc. (2) 61 (2000), no. 3, 789-806 https://doi.org/10.1112/S0024610700008759
12. 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
13. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228 https://doi.org/10.1002/cpa.3160280203

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