Energy finite solutions of elliptic equations on Riemannian manifolds
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We prove that for any continuous function f on the s-harmonic (1 < s < ∞) 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 E₁,E₂,...,El 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 Ei which vanish at the boundary ∂Ei for i = 1, 2,..., l.
Abstract
We prove that for any continuous function f on the s-harmonic (1 < s < ∞) 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 E₁,E₂,...,El 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 Ei which vanish at the boundary ∂Ei for i = 1, 2,..., l.
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