On the singular planar Plateau problem

Given any Γ=γ(S¹)², image of a Lipschitz curve γ:S¹→ R², not necessarily injective, we provide an explicit formula for computing the value of \[ A(γ):=inf\{. ∫B_1(0)|det(∇ u)| d x \ | \ u=γ on S^1\}, \] where the infimum is evaluated among all Lipschitz maps u:B₁(0)→ R² having boundary datum γ. This coincides with the area of a minimal disk spanning Γ, i.e., a solution of the Plateau problem of disk type for the oriented contour Γ. The novelty of the results relies in the fact that we do not assume the curve γ to be injective and our formula allows for any kind of self-intersections