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A Sharp Bound for the Growth of Minimal Graphs

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Abstract

We consider minimal graphs \(u = u(x,y) > 0\) over unbounded domains \(D \subset R^2\) bounded by a Jordan arc \(\gamma \) on which \(u = 0\). We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector

$$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$

then the rate of growth is at most \(r^{\pi /\lambda }\).

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References

  1. Drasin, D.: On Nevanlinna’s inverse problem. Complex Var. 37, 123–143 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Drasin, D., Weitsman, A.: Meromorphic functions with large sums of deficiencies. Adv. Math. 15, 93–126 (1974)

    Article  MathSciNet  Google Scholar 

  3. Duren, P.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  4. Hwang, J.-F.: Phragmén Lindelöf theorem for the minimal surface equation. Proc. Am. Math. Soc. 104, 825–828 (1988)

    MATH  Google Scholar 

  5. Jenkins, H., Serrin, J.: Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation. Arch. Rat. Mech. Anal., 21, 321–342 (1965/66)

  6. Lundberg, E., Weitsman, A.: On the growth of solutions to the minimal surface equation over domains containing a half plane. Calc. Var. Part. Differ. Equ. 54, 3385–3395 (2015)

    Article  Google Scholar 

  7. Miklyukov, V.: Some singularities in the behavior of solutions of equations of minimal surface type in unbounded domains. Math. USSR Sbornik 44, 61–73 (1983)

    Article  Google Scholar 

  8. Nitsche, J.C.C.: On new results in the theory of minimal surfaces. Bull. Am. Mat. Soc. 71, 195–270 (1965)

    Article  MathSciNet  Google Scholar 

  9. Osserman, R.: A Survey of Minimal Surfaces. Dover Publications Inc, New York (1986)

    MATH  Google Scholar 

  10. Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen Co. Ltd, Tokyo (1959)

    MATH  Google Scholar 

  11. Weitsman, A.: On the growth of minimal graphs. Indiana Univ. Math. J. 54, 617–625 (2005)

    Article  MathSciNet  Google Scholar 

  12. Weitsman, A.: Growth of solutions to the minimal surface equation over domains in a half plane. Commun. Anal. Geometry 13, 1077–1087 (2005)

    Article  MathSciNet  Google Scholar 

  13. Weitsman, A.: Level curves of minimal graphs. Arxiv:2008.10197, To appear in Comm. Anal. Geom. (2022)

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Authors and Affiliations

  1. Purdue University, West Lafayette, USA

    Allen Weitsman

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  1. Allen Weitsman
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Correspondence to Allen Weitsman.

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Communicated by Aimo Hinkkanen.

Dedicated with gratitude to the memory of Walter Hayman.

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Weitsman, A. A Sharp Bound for the Growth of Minimal Graphs. Comput. Methods Funct. Theory 21, 905–914 (2021). https://doi.org/10.1007/s40315-021-00417-1

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  • DOI: https://doi.org/10.1007/s40315-021-00417-1

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