A uniqueness theorem for harmonic functions on the upper-half plane

Author:
Biao Ou

Journal:
Conform. Geom. Dyn. **4** (2000), 120-125

MSC (2000):
Primary 53A30, 35J05, 30C15

DOI:
https://doi.org/10.1090/S1088-4173-00-00067-9

Published electronically:
December 15, 2000

MathSciNet review:
1799653

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider harmonic functions on the upper-half plane satisfying the boundary condition and the constraint . We prove that all such functions are of form (1.2) below.

**[A]**Aubin, T.,*Nonlinear analysis on manifolds, Monge-Ampere equations*, Springer-Verlag, New York, 1982. MR**85j:58002****[ABR]**Axler, S., Bourdon, P., and Ramey, W.,*Harmonic function theory*, Springer-Verlag, New York, 1992. MR**93f:31001****[CY]**Chang, A. and Yang, P.,*On uniqueness of solutions of**th order differential equations in conformal geometry*, Math. Res. Lett. 4 (1997), 91 - 102. MR**97m:58204****[CL]**Chen, W.-X. and Li, C.-M.,*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J. 63 (1991), 615 - 622. MR**93e:35009****[CSF]**Chipot, M., Shafrir, I., and Fila, M.,*On the solutions to some elliptic equations with nonlinear Neumann boundary conditions*, Adv. Differential Equations 1 (1996), 91 - 110.**[E]**Escobar, J.,*Sharp constant in a Sobolev trace inequality*, Indiana J. Math. 37 (1988), 687 - 698. MR**90a:46071****[K]**Krantz, S.,*Complex analysis: the geometric viewpoint*, Mathematical Association of America, Washington DC, 1990. MR**92a:30026****[LZ]**Li, Y.-Y. and Zhu, M.-J.,*Uniqueness theorems through the method of moving spheres*, Duke Math. J. 80 (1995), 383 - 417. MR**96k:35061****[O]**Ou, B.,*Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition*, Differential Integral Equations 9 (5) (1996), 1157 - 1164. MR**97i:35054**

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Additional Information

**Biao Ou**

Affiliation:
Department of Mathematics, University of Toledo, Toledo, Ohio 43606

Email:
bou@math.utoledo.edu

DOI:
https://doi.org/10.1090/S1088-4173-00-00067-9

Keywords:
Harmonic,
analytic,
the Picard theorems

Received by editor(s):
August 14, 2000

Published electronically:
December 15, 2000

Article copyright:
© Copyright 2000
American Mathematical Society