On two-dimensional parametric variational problems

Abstract.

Let $\mathcal{F}(X)$ be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that $\Gamma$ is a closed rectifiable Jordan curve in $\mathbb{R}^{3}$. We then prove that there is a conformally parametrized minimizer of $\mathcal{F}$ in the class $\mathcal{C}(\Gamma)$ of surfaces $X \in H^{1,2}(B, \mathbb{R}^{3}) \cap C^{0}(\overline{B}, \mathbb{R}^{3})$ of the type of the disk B which are bounded by $\Gamma$. An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on $\epsilon$-conformal mappings. In addition we show that the minimizer of $\mathcal{F}$ is Hölder continuous in B, and even in $\overline{B}$ if $\Gamma$ satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one.