the Lagrangian F(x, z) of which is positive definite and at least semi-elliptic. It turns out that there always exists a conformally para-me-trized minimizer. Any such minimizer X is seen to be Hölder-continuous in the parameter domain B and continuous up to its boundary. If F possesses a perfect dominance function G of class C², we can establish higher regularity of X in the interior. In fact, we prove X ⊆ H_loc^2,2(B, ℝⁿ) ∩ C^1,σ(B, ℝⁿ) for some σ > 0. Finally, we discuss the existence of perfect dominance functions.

BIBLIOGRAPHY

1 Acerbi, E.; Fusco, N. Semicontinuity problems in the calculus of variations. Arch Rational Mech Anal 86 (1984), no. 2, 125–145.
10.1007/BF00275731
Web of Science® Google Scholar
2 Courant, R. Dirichlet's principle, conformal mapping, and minimal surfaces. Interscience, New York, 1950.
Web of Science® Google Scholar
3 Dacorogna, B. Direct methods in the calculus of variations. Applied Mathematical Sciences, 78. Springer, Berlin, 1989.
10.1007/978-3-642-51440-1
Web of Science® Google Scholar
4 Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O. Minimal surfaces. I. Boundary value problems. II. Boundary regularity. Grundlehren der Mathematischen Wissenschaften, 295, 296. Springer, Berlin, 1992.
Google Scholar
5 Grüter, M. Conformally invariant variational integrals and the removability of isolated singularities. Manuscripta Math 47 (1984), no. 1–3, 85–104.
10.1007/BF01174588
Web of Science® Google Scholar
6 Hildebrandt, S.; Kaul, H. Two-dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold. Comm Pure Appl Math 25 (1972), 187–223.
10.1002/cpa.3160250208
Web of Science® Google Scholar
7 Hildebrandt, S.; von der Mosel, H. On two-dimensional parametric variational problems. Calc Var Partial Differential Equations 9 (1999), no. 3, 249–267.
10.1007/s005260050140
Web of Science® Google Scholar
8 Hildebrandt, S.; von der Mosel, H. Dominance functions for parametric Lagrangians. Geometric analysis and nonlinear partial differential equations. Edited by S. Hildebrandt and H. Karcher. Springer, Berlin–Heidelberg–New York, 2002.
Google Scholar
9 Hildebrandt, S.; von der Mosel, H. Plateau's problem for parametric double integrals: II. Regularity at the boundary. Preprint 38, SFB 611, Bonn University, 2002.
Google Scholar
Hildebrandt, S.; von der Mosel, H. J Reine Angew Math, to appear.
Google Scholar
10 Morrey, C. B. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J Math 2 (1952), 25–53.
10.2140/pjm.1952.2.25
Google Scholar
11 Morrey, C. B. The parametric variational problem for double integrals. Comm Pure Appl Math 14 (1961), 569–575.
10.1002/cpa.3160140328
Web of Science® Google Scholar
12 Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966.
10.1007/978-3-540-69952-1
Google Scholar
13 Nirenberg, L. Remarks on strongly elliptic partial differential equations. Comm Pure Appl Math 8 (1955), 649–675.
10.1002/cpa.3160080414
Web of Science® Google Scholar
14 Reshetnyak, Y. G. New proof of the theorem on existence of an absolute minimum for two-dimensional variational problems in parametric form. Sibirsk Mat Zh 3 (1962), 744–768.
Google Scholar

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Volume56, Issue7

First of Two Special Issues Dedicated to the Memory of Jürgen K. Moser

July 2003

Pages 926-955

Plateau's problem for parametric double integrals: I. Existence and regularity in the interior^†

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Plateau's problem for parametric double integrals: I. Existence and regularity in the interior^†