多重調和関数論に於けるリュービルの定理

概要

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The primary purpose of this paper is to propose and prove by Fourier expansion method a theorem of Liouville type for polyharmonic functions which is a direct extension of our former result given for harmonic functions which contains the classical Liouville theorem. The important tool for the above proof is the so-called Almansi decomposition of polyharmonic functions, the necessary and sufficient conditions for the possibility of which is explained in detail from our point of view just for the sake of convenience. The uniqueness of the decomposition is also proved for a more general family of functions including polyharmonic functions. Our theorem is compared with those of Armitage, Mizuta, and Futamura-Kishi-Mizuta, among others, relevant to Liouville type assertions for polyharmonic functions, and the relations between them are clarified. Finally we give a detailed proof of a general theorem of Liouville type not only for polyharmonic functions but also for a much more wide family of functions including polyharmonic functions. This result contains our main result mentioned above.
大同工業大学の論文