Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

  • Kim, Seok-Woo (Department of Mathematics Education Konkuk University) ;
  • Lee, Yong-Hah (Department of Mathematics Education Ewha Womans University)
  • Published : 2008.05.31
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Abstract

We prove that for any continuous function f on the s-harmonic (1{\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 ${\partial}E_{\iota}\;for\;{\iota}=1,2,...,{\iota}$

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References

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