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Completeness and existence of bounded biharmonic functions on a riemannian manifold

Leo Sario — 1974

Annales de l'institut Fourier

A.S. Galbraith has communicated to us the following intriguing problem: does the completeness of a manifold imply, or is it implied by, the emptiness of the class H 2 B of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry H 2 B -functions, and one might suspect that this is always the case. We shall show, however, that there do exist complete manifolds of any dimension that carry...

Behavior of biharmonic functions on Wiener's and Royden's compactifications

Y. K. KwonLeo SarioBertram Walsh — 1971

Annales de l'institut Fourier

Let R be a smooth Riemannian manifold of finite volume, Δ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of R are found, and for biharmonic functions (those for which Δ Δ u = 0 ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

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