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

HTML articles powered by AMS MathViewer

- by Biao Ou PDF
- Conform. Geom. Dyn.
**4**(2000), 120-125 Request permission

## Abstract:

Consider harmonic functions on the upper-half plane $R^{2}_{+}\! = \{(x,y)|\;y > 0 \}$ satisfying the boundary condition $u_{y}=-\exp (u)$ and the constraint $\int _{R^{2}_{+}}\exp (2u) < \infty$. We prove that all such functions are of form (1.2) below.## References

- Thierry Aubin,
*Nonlinear analysis on manifolds. Monge-Ampère equations*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR**681859**, DOI 10.1007/978-1-4612-5734-9 - Sheldon Axler, Paul Bourdon, and Wade Ramey,
*Harmonic function theory*, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR**1184139**, DOI 10.1007/b97238 - Sun-Yung A. Chang and Paul C. Yang,
*On uniqueness of solutions of $n$th order differential equations in conformal geometry*, Math. Res. Lett.**4**(1997), no. 1, 91–102. MR**1432813**, DOI 10.4310/MRL.1997.v4.n1.a9 - Wen Xiong Chen and Congming Li,
*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J.**63**(1991), no. 3, 615–622. MR**1121147**, DOI 10.1215/S0012-7094-91-06325-8
[CSF]4 Chipot, M., Shafrir, I., and Fila, M., - José F. Escobar,
*Sharp constant in a Sobolev trace inequality*, Indiana Univ. Math. J.**37**(1988), no. 3, 687–698. MR**962929**, DOI 10.1512/iumj.1988.37.37033 - Steven G. Krantz,
*Complex analysis: the geometric viewpoint*, Carus Mathematical Monographs, vol. 23, Mathematical Association of America, Washington, DC, 1990. MR**1074176** - Yanyan Li and Meijun Zhu,
*Uniqueness theorems through the method of moving spheres*, Duke Math. J.**80**(1995), no. 2, 383–417. MR**1369398**, DOI 10.1215/S0012-7094-95-08016-8 - Biao Ou,
*Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition*, Differential Integral Equations**9**(1996), no. 5, 1157–1164. MR**1392100**

*On the solutions to some elliptic equations with nonlinear Neumann boundary conditions*, Adv. Differential Equations 1 (1996), 91 - 110.

## Additional Information

**Biao Ou**- Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606
- Email: bou@math.utoledo.edu
- Received by editor(s): August 14, 2000
- Published electronically: December 15, 2000
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1799653