On the singular planar Plateau problem

Given any $\mathrm{\Gamma }=\gamma ({\mathbb{S}}^1)\subset {\mathbb{R}}^2$, image of a Lipschitz curve $\gamma :{\mathbb{S}}^1\to {\mathbb{R}}^2$, not necessarily injective, we provide an explicit formula for computing the value of $$\mathcal{A}(\gamma ):=\inf \{{\int }_{B_1(0)}\mathrm{\mid }\mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{\nabla }u)\mathrm{\mid }\mathrm{d}x\mid u=\gamma \text{ on }{\mathbb{S}}^1\},$$ where the infimum is evaluated among all Lipschitz maps $u:B_1(0)\to {\mathbb{R}}^2$ having boundary datum $\gamma$. This coincides with the area of a minimal disk spanning $\mathrm{\Gamma }$, i.e., a solution of the Plateau problem of disk type for the oriented contour $\mathrm{\Gamma }$. The novelty of the results relies in the fact that we do not assume the curve $\gamma$ to be injective and our formula allows for any kind of self-intersections