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On integral representations for the Neumann operator.

D H Phong1

  • 1School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1979
PubMed
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Researchers derived an approximate formula for the Neumann operator kernel on pseudo-convex domains. This work introduces novel singular integral operators with unique smoothing and boundary properties.

Area of Science:

  • Complex analysis
  • Partial differential equations
  • Geometric analysis

Background:

  • The Neumann operator is crucial in solving boundary value problems.
  • Understanding its kernel is key for analyzing function spaces on domains.
  • Strongly pseudo-convex domains present unique analytical challenges.

Purpose of the Study:

  • To derive an approximate formula for the kernel of the Neumann operator (N).
  • To investigate properties of novel singular integral operators on strongly pseudo-convex domains.

Main Methods:

  • Derivation of an approximate formula for the Neumann operator kernel.
  • Analysis of singular integral operators with specific kernel structures.

Main Results:

Related Experiment Videos

  • An approximate formula for the Neumann operator kernel was established.
  • A new class of singular integral operators was identified.
  • These operators exhibit infinite smoothing in the interior and preserve singular supports at the boundary.
  • The kernels are characterized as products of isotropic and parabolic kernels.

Conclusions:

  • The derived formula provides a new tool for studying the Neumann operator.
  • The newly identified singular integral operators offer unique analytical properties relevant to boundary value problems on pseudo-convex domains.