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<p>In this paper, we utilize the technique of Laurent series to establish the Almansi decomposition theorem for polyharmonic functions on annular domains. We generalize the classical Almansi decomposition theorem to cases where it does not hold, and derive explicit decomposition formulae for various cases. Furthermore, we investigate the uniqueness of decomposition and provide all non-trivial decompositions of the zero function. Finally, we apply the established Almansi decomposition to solve the Dirichlet problem for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta ^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Δ</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> on annular domains.</p>

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Almansi Decomposition of Polyharmonic Functions on Annular Domains and Its Applications

  • 期刊论文

  • 发表日期：

    2025年12月20日

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  • Huafeng Zeng

摘要

In this paper, we utilize the technique of Laurent series to establish the Almansi decomposition theorem for polyharmonic functions on annular domains. We generalize the classical Almansi decomposition theorem to cases where it does not hold, and derive explicit decomposition formulae for various cases. Furthermore, we investigate the uniqueness of decomposition and provide all non-trivial decompositions of the zero function. Finally, we apply the established Almansi decomposition to solve the Dirichlet problem for \(\Delta ^{m}\) Δ m on annular domains.