Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The Dirichlet problem on quadratic surfaces

Authors: Sheldon Axler, Pamela Gorkin and Karl Voss
Journal: Math. Comp. 73 (2004), 637-651
MSC (2000): Primary 31B05, 31B20
Published electronically: June 10, 2003
MathSciNet review: 2031398
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in $\mathbf{R}^n$ such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in $\mathbf{R}^n$ can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

References [Enhancements On Off] (What's this?)

  • 1. Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, second edition, Graduate Texts in Mathematics, Vol. 137, Springer-Verlag (2001). MR 2001j:31001
  • 2. Sheldon Axler and Wade Ramey, Harmonic polynomials and Dirichlet-type problems, Proc. Amer. Math. Soc. 123 (1995), 3765-3773. MR 96b:31003
  • 3. John A. Baker, The Dirichlet problem for ellipsoids, Amer. Math. Monthly 106 (1999), 829-834. MR 2000j:35046
  • 4. Dmitry Khavinson and Harold S. Shapiro, Dirichlet's problem when the data is an entire function, Bull. London Math. Soc. 24 (1992), 456-468. MR 94d:35005
  • 5. Winfried Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, Vol. 270, Springer-Verlag (1985). MR 86k:11022
  • 6. Harold S. Shapiro, An algebraic theorem of E. Fischer, and the holomorphic Goursat problem, Bull. London Math. Soc. 21 (1989), 513-537. MR 90m:35008
  • 7. D. Siegel and E. O. Talvila, Uniqueness for the $n$-dimensional half space Dirichlet problem, Pacific J. Math. 175 (1996), 571-587. MR 98a:35020

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 31B05, 31B20

Retrieve articles in all journals with MSC (2000): 31B05, 31B20

Additional Information

Sheldon Axler
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132

Pamela Gorkin
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Karl Voss
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Keywords: Laplacian, Dirichlet problem, harmonic, ellipsoid, polynomial, quadratic surface
Received by editor(s): November 11, 2002
Published electronically: June 10, 2003
Additional Notes: The first author was supported in part by the National Science Foundation
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society