On convexity of level curves of harmonic functions in the hyperbolic plane

Author:
M. Papadimitrakis

Journal:
Proc. Amer. Math. Soc. **114** (1992), 695-698

MSC:
Primary 31A05; Secondary 30F45, 52A55

MathSciNet review:
1086339

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if two level curves of a harmonic function are convex in the hyperbolic disc then all intermediate level curves are also convex.

**[1]**Luis A. Caffarelli and Avner Friedman,*Convexity of solutions of semilinear elliptic equations*, Duke Math. J.**52**(1985), no. 2, 431–456. MR**792181**, 10.1215/S0012-7094-85-05221-4**[2]**Luis A. Caffarelli and Joel Spruck,*Convexity properties of solutions to some classical variational problems*, Comm. Partial Differential Equations**7**(1982), no. 11, 1337–1379. MR**678504**, 10.1080/03605308208820254**[3]**R. M. Gabriel,*An extended principle of the maximum for harmonic functions in 3-dimensions*, J. London Math. Soc.**30**(1955), 388–401. MR**0072959****[4]**R. M. Gabriel,*A result concerning convex level surfaces of 3-dimensional harmonic functions*, J. London Math. Soc.**32**(1957), 286–294. MR**0090662****[5]**R. M. Gabriel,*Further results concerning the level surfaces of the Green’s function for a 3-dimensional convex domain. II*, J. London Math. Soc.**32**(1957), 303–306. MR**0090664****[6]**J. J. Gergen,*Note on the Green Function of a Star-Shaped Three Dimensional Region*, Amer. J. Math.**53**(1931), no. 4, 746–752. MR**1506851**, 10.2307/2371223**[7]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[8]**Nicholas J. Korevaar,*Convex solutions to nonlinear elliptic and parabolic boundary value problems*, Indiana Univ. Math. J.**32**(1983), no. 4, 603–614. MR**703287**, 10.1512/iumj.1983.32.32042**[9]**Nicholas J. Korevaar and John L. Lewis,*Convex solutions of certain elliptic equations have constant rank Hessians*, Arch. Rational Mech. Anal.**97**(1987), no. 1, 19–32. MR**856307**, 10.1007/BF00279844**[10]**John L. Lewis,*Capacitary functions in convex rings*, Arch. Rational Mech. Anal.**66**(1977), no. 3, 201–224. MR**0477094****[11]**Jean-Pierre Rosay and Walter Rudin,*A maximum principle for sums of subharmonic functions, and the convexity of level sets*, Michigan Math. J.**36**(1989), no. 1, 95–111. MR**989939**, 10.1307/mmj/1029003884

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1086339-3

Article copyright:
© Copyright 1992
American Mathematical Society