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Free Boundary Problems for Surfaces of Constant Mean Curvature

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Book cover Variational Methods for Free Surface Interfaces

Abstract

This survey describes a new existence result [21] for (disk-type) surfaces of prescribed constant mean curvature with free boundaries, and relates this result to some other well-known variational problems arising in differential geometry.

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References

  1. H. Brezis and J.-M. Coron, Multiple solutions of H-systems and Rellich’s conjecture, Comm. Pure Appl. Math. 37 (1984), 149–187.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Courant, Dirichlet’s principle, conformal mapping and minimal surfaces, Interscience, New York, 1950.

    MATH  Google Scholar 

  3. G. Dziuk, C 2 -regularity for partially free minimal surfaces, Math. Z. 189 (1985), 71–79.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Grüter, S. Hildebrandt, and J.C.C. Nitsche, Regularity for stationary surfaces of constant mean curvature with free boundaries, Acta Math. (in press).

    Google Scholar 

  5. M. Grüter and J. Jost, On embedded minimal discs in convex bodies (Preprint).

    Google Scholar 

  6. S. Hildebrandt, On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970), 97–114.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Hildebrandt and J.C.C. Nitsche, Minimal surfaces with free boundaries, Acta Math. 143 (1979), 251–272.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Jost, Existence results for embedded minimal surfaces of controlled topological type (Preprint).

    Google Scholar 

  9. L. Ljusternik and L. Schnirelman, Existence de trois géodésiques fermées sur toute surface de genre o, C.R. Acad. Sci. Paris 188 (1929), 534–536.

    Google Scholar 

  10. U. Massari, Esistenza e regolaritá delle ipersuperfici di curvature media assegnata inn, Arch. Rat. Mech. Anal. 55 (1974), 357–382.

    Article  MathSciNet  MATH  Google Scholar 

  11. J.C.C. Nitsche, Stationary partitioning of convex bodies, Arch. Rat. Mech. Anal. 89 (1985), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  12. R.S. Palais and S. Smale, A generalized Morse theory, Bull. AMS 70 (1964), 165–171.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. H.A. Schwarz, Gesammelte Mathematische Abhandlungen, Band I, Springer, Berlin, 1890.

    Google Scholar 

  15. B. Smyth, Stationary minimal surfaces with boundary on a simplex, Inv. Math. 76 (1984), 411–420.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Steffen, On the nonuniqueness of surfaces with prescribed constant mean curvature spanning a given contour, Arch. Rat. Mech. Anal. 94 (1986), 101–122.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Struwe, Nonuniqueness in the Plateau problem for surfaces of constant mean curvature, Arch. Rat. Mech. Anal. 93 (1986), 135–157.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Struwe, Large H-surfaces via the mountain-pass-lemma, Math. Ann. 270 (1985), 441–459.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comm. Math. Helv. 60 (1985), 558–581.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Struwe, The existence of surfaces of constant mean curvature with free boundaries (Preprint).

    Google Scholar 

  22. J.E. Taylor, Boundary regularity for solutions to various capillarity and free boundary problems, Comm. PDE 2 (1977), 323–357.

    Article  MATH  Google Scholar 

  23. P. Tolksdorf, A parametric variational principle for minimal surfaces of varying topological type, J. Reine Angew. Math. 354 (1984), 16–49.

    Article  MathSciNet  MATH  Google Scholar 

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© 1987 Springer-Verlag New York Inc.

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Struwe, M. (1987). Free Boundary Problems for Surfaces of Constant Mean Curvature. In: Concus, P., Finn, R. (eds) Variational Methods for Free Surface Interfaces. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4656-5_6

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  • DOI: https://doi.org/10.1007/978-1-4612-4656-5_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-9101-5

  • Online ISBN: 978-1-4612-4656-5

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