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A class of functions containing polyharmonic functions in ℝⁿ

V. Anandam, M. Damlakhi (2003)

Annales Polonici Mathematici

Some properties of the functions of the form v ( x ) = i = 0 m | x | i h i ( x ) in ℝⁿ, n ≥ 2, where each h i is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

Axiomatique des fonctions biharmoniques. I

Emmanuel P. Smyrnelis (1975)

Annales de l'institut Fourier

Les théories axiomatiques existantes de fonctions harmoniques ne s’appliquent pas à des équations simples d’ordre > 2 , comme l’équation biharmonique Δ 2 u = Δ ( Δ u ) = 0 ou le système équivalent Δ u 1 = - u 2 , Δ u 2 = 0 .On développe donc ici, au moyen d’un faisceau de couples convenables de fonctions ( u 1 , u 2 ) une approche axiomatique locale applicable à des équations du type L 2 ( L 1 u ) = 0 , où L j ( j = 1 , 2 ) est un opérateur linéaire du second ordre elliptique ou parabolique. Deux axiomatiques harmoniques lui sont associées. On traite, dans ce cadre, le problème (généralisé)...

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

Y. K. Kwon, Leo Sario, Bertram 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.

Biharmonic Green domains in a Riemannian manifold

Sadoon Ibrahim Othman, Victor Anandam (2003)

Commentationes Mathematicae Universitatis Carolinae

Let R be a Riemannian manifold without a biharmonic Green function defined on it and Ω a domain in R . A necessary and sufficient condition is given for the existence of a biharmonic Green function on Ω .

Biharmonic morphisms

Mustapha Chadli, Mohamed El Kadiri, Sabah Haddad (2005)

Commentationes Mathematicae Universitatis Carolinae

Let ( X , ) and ( X ' , ' ) be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from ( X , ) to ( X ' , ' ) is a continuous map from X to X ' which preserves the biharmonic structures of X and X ' . In the present work we study this notion and characterize in some cases the biharmonic morphisms between X and X ' in terms of harmonic morphisms between the harmonic spaces associated with ( X , ) and ( X ' , ' ) and the coupling kernels of them.

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...

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