Energy finite solutions of elliptic equations on Riemannian manifolds

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 M-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. ©2008 The Korean Mathematical Society.