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Duality by reproducing kernels

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  • A. Shlapunov
  • N. Tarkhanov

Abstract

Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X . Write 𝒮 A ( 𝒟 ) for the space of solutions of the system A u = 0 in a domain 𝒟 ⋐ X . Using reproducing kernels related to various Hilbert structures on subspaces of 𝒮 A ( 𝒟 ) , we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of 𝒟 , we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂ ¯ -Neumann problem. The duality itself takes place only for those domains 𝒟 which possess certain convexity properties with respect to A .

Suggested Citation

  • A. Shlapunov & N. Tarkhanov, 2003. "Duality by reproducing kernels," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-69, January.
    • Handle: RePEc:hin:jijmms:387058
    DOI: 10.1155/S0161171203206037
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