The asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries

Frank Müller

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 529-559
  • ISSN: 0391-173X

Abstract

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We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.

How to cite

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Müller, Frank. "The asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 529-559. <http://eudml.org/doc/272297>.

@article{Müller2007,
abstract = {We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.},
author = {Müller, Frank},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {partially free boundary alue problem; Jordan arc; stationary surfaces; prescribed mean curvature; partially free boundary value problem; Jordan arc},
language = {eng},
number = {4},
pages = {529-559},
publisher = {Scuola Normale Superiore, Pisa},
title = {The asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries},
url = {http://eudml.org/doc/272297},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Müller, Frank
TI - The asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 529
EP - 559
AB - We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.
LA - eng
KW - partially free boundary alue problem; Jordan arc; stationary surfaces; prescribed mean curvature; partially free boundary value problem; Jordan arc
UR - http://eudml.org/doc/272297
ER -

References

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  1. [1] U. Dierkes, S. Hildebrandt, A. Küster and O. Wohlrab, “Minimal Surfaces I, II", Grundlehren Math. Wiss. Vol. 295, 296. Springer, Berlin Heidelberg New York, 1992. Zbl0777.53012MR1215267
  2. [2] G. Dziuk, Über quasilineare elliptische Systeme mit isothermen Parametern an Ecken der Randkurve, Analysis1 (1981), 63–81. Zbl0485.35012MR623643
  3. [3] G. Dziuk, On the boundary behaviour of partially free minimal surfaces, Manuscripta Math.35 (1981), 105–123. Zbl0477.49024MR627928
  4. [4] K. Goldhorn and S. Hildebrandt, Zum Randverhalten der Lösungen gewisser zweidimensionaler Variationsprobleme mit freien Randbedingungen, Math. Z.118 (1970), 241–253. Zbl0202.20502MR279699
  5. [5] P. Hartman and A. Wintner, On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math.75 (1953), 449–476. Zbl0052.32201MR58082
  6. [6] E. Heinz, Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern, Math. Z.113 (1970), 99–105. Zbl0176.41004MR262683
  7. [7] E. Heinz, Über die analytische Abhängigkeit der Lösungen eines linearen elliptischen Randwertproblems von Parametern, Nachr. Akad. Wiss. Göttingen II, Math.-Phys. Kl. (1979), 1–20. Zbl0436.35010MR568799
  8. [8] S. Hildebrandt, Einige Bemerkungen über Flächen beschränkter mittlerer Krümmung,- Math. Z.115 (1970), 169-178. Zbl0185.50201MR266115
  9. [9] F. Müller, A priori bounds for H -surfaces in a partially free configuration, Analysis26 (2006), 471–489. Zbl1133.53006MR2329588
  10. [10] F. Müller, Investigations on the regularity of surfaces with prescribed mean curvature and partially free boundaries, Habilitationsschrift, BTU Cottbus (2006). 
  11. [11] F. Müller, On the regularity of H -surfaces with free boundaries on a smooth support manifold, BTU Cottbus (2007), preprint. Zbl1161.53011
  12. [12] F. Müller, Growth estimates for the gradient of an H -surface near singular points of the boundary configuration, Z. Anal. Anwendungen, to appear. Zbl1167.53012MR2469718
  13. [13] J. C. C. Nitsche, Minimal surfaces with partially free boundary. Least area property and Hölder continuity for boundaries satisfying a chord-arc condition, Arch. Ration. Mech. Anal. 39 (1970), 131–145. Zbl0209.41602MR266068
  14. [14] F. Sauvigny, “Partial Differential Equations 1: Foundations and Integral Representations", Springer, Berlin Heidelberg, 2006. Zbl1198.35001MR2254749
  15. [15] F. Sauvigny, “Partial Differential Equations 2: Functional Analytic Methods", Springer, Berlin Heidelberg, 2006. Zbl1198.35002MR2254750

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