The Neumann problem on ellipsoids

Original Research

Abstract

The Neumann problem on an ellipsoid in \(\mathbf {R}^n\) asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a normalized polynomial (a polynomial divided by the norm of the normal vector arising from the definition of the ellipsoid). Specifically, we give a necessary and sufficient condition for a solution to exist, and we show that if a solution exists then it is a polynomial whose degree is at most the degree of the polynomial giving the specified function. Furthermore, we give an algorithm for computing this solution. We also solve the corresponding generalized Neumann problem and give an algorithm for computing its solution.

Keywords

Neumann problem Ellipsoid Harmonic function 

Mathematics Subject Classification

Primary 31B05 31B20 

Notes

Acknowledgements

Peter J. Shin was supported by National Institutes of Health Grant Number P41-EB013598.

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Copyright information

© Korean Society for Computational and Applied Mathematics 2017

Authors and Affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Department of Radiology and Biomedical ImagingUniversity of California, San FranciscoSan FranciscoUSA

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