A uniqueness theorem for harmonic functions on the upper-half plane
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- by Biao Ou
- Conform. Geom. Dyn. 4 (2000), 120-125
- DOI: https://doi.org/10.1090/S1088-4173-00-00067-9
- Published electronically: December 15, 2000
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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., On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Adv. Differential Equations 1 (1996), 91 - 110.
- 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
Bibliographic 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