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

We study Plateau's problem for two‐dimensional parametric integrals$${\cal F}(X) := \int_{B} F(X, X_{u} \wedge X_{v}) \,du \,dv,$$ 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 ⊆ H loc 2,2 ( B , ℝ n ) ∩ C 1,σ ( B , ℝ n ) for some σ > 0. Finally, we discuss the existence of perfect dominance functions.