Skip to main content
SpringerLink
Log in

A Lower Bound on the Growth of Minimal Graphs

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

We show that for minimal graphs in \(R^3\) having 0 boundary values over simpy connected domains, the maximum over circles of radius r must be at least of the order \(r^{1/2}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Not applicable

References

  1. Ahlfors, L.: Lectures on quasiconformal mappings. In: Van Nostrand Mathematical Studies (1966)

  2. Duren, P.: Harmonic mappings in the plane. In: Cambridge Tracts in Mathematics (2004)

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

    Article  Google Scholar 

  4. Mikljukov, 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 

  5. Nevanlinna, R.: Analytic Functions. Springer, Berlin (1970)

    Book  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Weitsman, A.: A sharp bound for the growth of minimal graphs. Comput. Methods Funct. Theory 21, 905–914 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1395, USA

    Allen Weitsman

Authors
  1. Allen Weitsman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Allen Weitsman.

Additional information

Communicated by Gaven Martin.

Dedicated to the memory of Peter Duren with gratitude for his contributions to classical function theory.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Weitsman, A. A Lower Bound on the Growth of Minimal Graphs. Comput. Methods Funct. Theory (2024). https://doi.org/10.1007/s40315-024-00532-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40315-024-00532-9

Keywords

Mathematics Subject Classification

Search

Navigation