On the singular planar Plateau problem

Given any = γ(S1) ⊂ R2, image of a Lipschitz curve γ : S1 → R2, not necessarily injective, we provide an explicit formula for computing the value of A(γ) := inf | det(∇u)|dx u = γ on S1 , B1(0) where the infimumiscomputedamongall Lipschitz maps u : B1(0) → R2 havingboundary datum γ. This coincides with the area of a minimal disk spanning , i.e., a solution of the Plateau problem of disk type. The novelty of the results relies in the fact that we do not assume the curve γ to be injective and our formula allows arbitrary self-intersections.