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

DOI:
https://doi.org/10.1090/S0025-5718-03-01574-6

Published electronically:
June 10, 2003

MathSciNet review:
2031398

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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 such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in 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.

**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 -dimensional half space Dirichlet problem*, Pacific J. Math.**175**(1996), 571-587. MR**98a:35020**

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Additional Information

**Sheldon Axler**

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

Email:
axler@sfsu.edu

**Pamela Gorkin**

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

Email:
pgorkin@bucknell.edu

**Karl Voss**

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

Email:
kvoss@bucknell.edu

DOI:
https://doi.org/10.1090/S0025-5718-03-01574-6

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