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MiS Preprint
88/2001
Plateaus' problem for parametric double integrals: I. Existence and regularity in the interior
Stefan Hildebrandt and Heiko von der Mosel
Abstract
We study Plateau's problem for two-dimensional parametric integrals $$F(X):= \int_B F(X,X_U \wedge X_v),dudv$$ 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 functionG of class $C^2$ we can establish higher regularity of X in the interior. In fact, we prove $X\in H^{2,2}_{loc} (B,^n)$ for some $\sigma > 0$ Finally we discuss the existence of perfect dominance functions.
