An Integral Equation and a General Existence Theorem for Harmonic Functions.
Completeness and existence of bounded biharmonic functions on a riemannian manifold
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 of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry -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...
Generators of the Space of Bounded Biharmonic Functions.
Dirichlet Finite Biharmonic Functions with Dirichlet Finite Laplacians.
Royden's Algebra on Riemannian Spaces.
Classification and Deformation of Riemannian Spaces.
A nonexistence test for biharmonic Green's functions of clamped bodies.
Existence of bounded biharmonic functions.
On Hadamard's problem for higher dimensions.
Manifolds with strong harmonic boundaries but without Green's functions of clamped bodies
Quasiharmonic -functions on riemannian manifolds
Riemannian manifolds with bounded Dirichlet finite polyharmonic functions
Behavior of biharmonic functions on Wiener's and Royden's compactifications
Let be a smooth Riemannian manifold of finite volume, its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of are found, and for biharmonic functions (those for which ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.
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