On the singular planar Plateau problem

Given any \(\Gamma =\gamma (\mathbb S^1)\subset \mathbb {R}^2\) $\mathrm{\Gamma }=\gamma ({\mathbb{S}}^1)\subset {\mathbb{R}}^2$ , image of a Lipschitz curve \(\gamma :\mathbb S^1\rightarrow \mathbb {R}^2\) $\gamma :{\mathbb{S}}^1\to {\mathbb{R}}^2$ , not necessarily injective, we provide an explicit formula for computing the value of \(\begin{aligned} \mathcal {A}(\gamma ):=\inf \left\{ \left. \int _{B_1(0)}|\det (\nabla u)| \textrm{d}x \ \right| \ u=\gamma \text { on }\mathbb S^1\right\} , \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{A}(\gamma ):=inf\left\{\left({\int }_{B_1(0)},|det(\mathrm{\nabla }u)|,\text{d},x,\right|,,u,=,\gamma ,,\text{on},,{\mathbb{S}}^1\right\},\end{array}$$ where the infimum is computed among all Lipschitz maps \(u:B_1(0)\rightarrow \mathbb {R}^2\) $u:B_1(0)\to {\mathbb{R}}^2$ having boundary datum \(\gamma \) $\gamma$ . This coincides with the area of a minimal disk spanning \(\Gamma \) $\mathrm{\Gamma }$ , 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 \(\gamma \) $\gamma$ to be injective and our formula allows arbitrary self-intersections.