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THE NEUMANN PROBLEM ON ELLIPSOIDS.

Sheldon Axler1, Peter J Shin2

  • 1Department of Mathematics, San Francisco State University, San Francisco, CA 94132 USA.

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Researchers solved the Neumann problem for harmonic functions on ellipsoids. They established conditions for polynomial solutions and developed algorithms for computation, including for the generalized Neumann problem.

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Area of Science:

  • Mathematical Physics
  • Partial Differential Equations
  • Harmonic Analysis

Background:

  • The Neumann problem involves finding a harmonic function within a domain based on its normal derivative on the boundary.
  • Ellipsoidal geometries present unique challenges in solving boundary value problems due to their complex geometry.

Purpose of the Study:

  • To solve the Neumann problem for harmonic functions on an ellipsoid in R^n.
  • To determine conditions for the existence of polynomial solutions when the normal derivative is a normalized polynomial.
  • To develop computational algorithms for these solutions.

Main Methods:

  • The study employs techniques from potential theory and the theory of harmonic functions.
  • It involves analyzing the properties of harmonic polynomials on ellipsoidal surfaces.
  • Methods include deriving necessary and sufficient conditions for solution existence.

Main Results:

  • A necessary and sufficient condition for the existence of a solution to the Neumann problem on an ellipsoid is established.
  • If a solution exists, it is proven to be a polynomial with a degree bounded by that of the specified normal derivative.
  • Algorithms for computing both the standard and generalized Neumann problem solutions are provided.

Conclusions:

  • The Neumann problem on an ellipsoid with a normalized polynomial normal derivative is solvable under specific conditions.
  • The solution, when it exists, is a polynomial, offering a constructive approach.
  • The developed algorithms provide practical tools for computing these solutions in mathematical physics and related fields.