Abstract
We extend a classical result of Radó and Kneser concerning uniqueness ofminimal surfaces bounded by a given closed Jordan curve Γ inℝ3 to the case of extremals for certain geometric variationalintegrals. Using standard elliptic PDE theory, this gives the existenceand uniqueness of embedded F-minimal surfaces for suitable boundarycurves that project simply onto the boundary of a plane convex domain.
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References
Clarenz, U.: Sätze über Extremalen zu parametrischen Funktionalen, Bonner Math. Schriften 322 (1999), 1–79.
Clarenz, U.: Enclosure theorems for extremals of elliptic parametric functionals, Calc. Var. 15(3) (2002), 313–324.
Clarenz, U. and von der Mosel, H.: On surfaces of prescribed F-mean curvature, Preprint 18, SFB 611, Universität Bonn, 2002.
Dierkes, U.: A geometric maximum principle, Plateau's problem for surfaces of prescribed mean curvature, and the two-dimensional analogue of the catenary, in: S. Hildebrandt and R. Leis (eds), Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, New York, 1988, pp. 116–141.
Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrab, O.: Minimal Surfaces I, Grundlehren Math. Wiss. 295, Springer, New York, 1992.
Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. of Math. 97(1) (1973), 275–305.
Gilbarg, D. and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, New York, 1977 (second edition, 1983).
Hildebrandt, S. and von der Mosel, H.: On two-dimensional parametric variational problems, Calc. Var. 9 (1999), 249–267.
Hildebrandt, S. and von der Mosel, H.: Plateau's problem for parametric double integrals. Part I: Existence and interior regularity, Preprint 745, SFB 256, Universität Bonn, 2001, Comm. Pure Appl. Math., to appear.
Hildebrandt, S. and von der Mosel, H.: Plateau's problem for parametric double integrals. Part II: Regularity at the boundary, Preprint 38, SFB 611, Universität Bonn, 2002.
Hartman, P. and Wintner, A.: On the local behaviour of solutions of nonparabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476.
Kneser, H.: Lösung der Aufgabe 41, Jahresbericht der DMV 35 (1926), 123–124.
Radó, T.: Some remarks on the problem of Plateau, Proc. Nat. Acad. Sci. USA 16 (1930), 242–248.
Radó, T.: Aufgabe 41, Jahresbericht der DMV 35 (1926), 49.
Räwer, K.: Stabile Extremalen parametrischer Doppelintegrale in ℝ3, Dissertation, Bonn, 1993.
White, B.: Existence and regularity of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds, J. Differential Geom. 33 (1991), 413–443.
Winklmann, S.: Verallgemeinerte Minimalflächen und Krümmungsflüsse, Diplomarbeit, Duisburg, 2002.
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Winklmann, S. Existence and Uniqueness of F-Minimal Surfaces. Annals of Global Analysis and Geometry 24, 269–277 (2003). https://doi.org/10.1023/A:1024765105599
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DOI: https://doi.org/10.1023/A:1024765105599