Skip to main content

Surfaces of Minimal Area Supported by a Given Body in ℝ3

  • Chapter
Variational Methods

Abstract

Existence theory in the “Plateau problem” is mostly concerned with surfaces spanning some given boundary configuration ([A], [Cou], [HiN], [GJ], [S], to quote a few).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alt, H.W.: Die Existenz einer Minimalflache mit freimen Rand vorgeschriebener Lange, Arch. Rat. Mech. Anal. 51 (1973), 304 – 320

    Article  MathSciNet  MATH  Google Scholar 

  2. Brezis, H., Coron,J.M.: Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92, 203 – 215 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezis, H., Coron, J.M., Lieb,E.H.: Harmonic maps with defects, Comm. Math. Phys. 107, 649 – 705 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Courant, R.: Dirichlet principle, conformal mappings and minimal surfaces, Interscience, New York, 1950.

    Google Scholar 

  5. De Giorgi E.: Problemi di superfici minime con ostacoli: forma non cartesiana, B.U.M.I., 8, 80-88 (1973).

    Google Scholar 

  6. Duzaar, F.: Variational inequalities and harmonic mappings, J. Reine Angewandte Math. 374, 39 – 60 (1987)

    MathSciNet  MATH  Google Scholar 

  7. Fuchs, M.: Variational inequalities for vector-valued functions with nonconvex obstacles, Analysis 5, 223 – 238 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Gruter, M., J.Jost: On embedded minimal disks in convex bodies, A.I.H.P. Analyse non lineaire, 3, 5, 1986, 345 – 390

    MathSciNet  Google Scholar 

  9. Hildebrandt, S.: On the regularity of solutions of two-imensional variational problems with obstructions, Comm. Pure Appl. Math. 25, 479-496 (1972)

    Google Scholar 

  10. Hildebrandt, S.: Boundary behaviour of minimal surfaces, Arc. Rat. Mech. Anal. 35, 47 – 82 (1969).

    Google Scholar 

  11. Hildebrandt, S., J.C.C. Nitsche: Minimal surfaces with free boundaries, Acta Matematica, 143 (1979) 251 – 272

    Article  MathSciNet  MATH  Google Scholar 

  12. Lions, P.L.: The concentration-Compactness principle in the Calculus of Variations: the limit case, Rev. Mat. Iberoamericana 1, 145-201 vol. 1 and 45 – 121 vol. 2 (1985)

    Article  Google Scholar 

  13. Musina R.: S2-type minimal surfaces enclosing many obstacles in R3, preprint. [MM] Mancini G., R. Musina: Surfaces of minimal area enclosing a given body in IR3 .

    Google Scholar 

  14. Micaleff M.J., J.D. Moore: Minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes, Ann. of Math., 198-227(1988).

    Google Scholar 

  15. Nitsche, J.C.C.: Vorlesungen uber minimalflachen, Springer, Berlin 1975

    Book  Google Scholar 

  16. Sacks J., K.Uhlenbeck: The existence of minimal immersions of 2-spheres, Ann. of Math., 113, 1-24 (1981)

    MathSciNet  Google Scholar 

  17. Struwe, M.: On a free boundary problem for minimal surfaces, Invent. Math., 75 (1984), 547 – 560

    MathSciNet  MATH  Google Scholar 

  18. Tomi, F.: Minimal surfaces and surfaces of prescribed mean curvature spanned over obstacles, Math. Ann. 190, 248 – 264 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wente, H.: Large Solutions to the Volume Constrained Plateau Problem, Arch. Rat. Mech. and Analysis, 75, 59 – 77 (1980).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer Science+Business Media New York

About this chapter

Cite this chapter

Mancini, G., Musina, R. (1990). Surfaces of Minimal Area Supported by a Given Body in ℝ3 . In: Berestycki, H., Coron, JM., Ekeland, I. (eds) Variational Methods. Progress in Nonlinear Differential Equations and Their Applications, vol 4. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-1080-9_22

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-1080-9_22

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4757-1082-3

  • Online ISBN: 978-1-4757-1080-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics