Abstract
We prove the existence of conformally paramaterized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class \( H_{loc}^{2,2} \cap C^{1,\alpha } ,0 < \alpha < 1 \), up to the free boundary under the assumption that there exists a perfect dominance function in the sense of Morrey.
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References
S. Hildebrandt and H. von der Mosel, On two-dimensional parametric variational problems Calc. Var. Partial Differ. Equ. 9 (1999), 249–267.
L. J. Lipkin, A free boundary problem for parametric integrals of the calculus of variations, Rend. Circ. Mat. Palermo, II. Ser. 17 (1968), 33–67.
C. B. Morrey, The parametric variational problem for double integrals, Commun. Pure Appl. Math. 14 (1961), 569–575.
R. Courant, Dirichlet’s principle, conformai mapping, and minimal surfaces, Interscience Publishers, New York, 1950.
U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal Surfaces, Vols. I, II. Grundlehren der math. Wissenschaften 295, 296, Springer, Berlin, 1992.
C. B. Morrey, Multiple integrals in the calculus of variations, Grundlehren der math. Wissenschaften 130, Springer, Berlin, 1966.
S. Hildebrandt and H. von der Mosel, Plateau’s problem for parametric double integrals: Part I. Existence and regularity in the interior, Preprint 745 (2001) SFB256, University of Bonn; Preprint 88 (2001) MPI Mathematics Leipzig. [To appear in Commun. Pure Appl. Math.]
S. Hildebrandt and H. von der Mosel, Dominance functions for parametric Lagrangians. [To appear]
S. Hildebrandt and H. von der Mosel, Plateau’s problem for parametric double integrals: Part II. Boundary regularity. [To appear]
E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984), 125–145.
S. Hildebrandt, Boundary behavior of minimal surfaces, Arch. Ration. Mech. Anal. 35 (1969), 47–82.
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Dedicated to Professor O. A. Ladyzhenskaya in admiration
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Hildebrandt, S., von der Mosel, H. (2002). The Partially Free Boundary Problem for Parametric Double Integrals. In: Birman, M.S., Hildebrandt, S., Solonnikov, V.A., Uraltseva, N.N. (eds) Nonlinear Problems in Mathematical Physics and Related Topics I. International Mathematical Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0777-2_9
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DOI: https://doi.org/10.1007/978-1-4615-0777-2_9
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