MiS Preprint Repository

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 function G of class $C^2$ we can establish higher regularity of X in the interior. In fact, we prove $X\in H_{loc}^{2,2}(B{,}^n)$ for some $\sigma >0$ Finally we discuss the existence of perfect dominance functions.