On the singular planar Plateau problem

Abstract

Given any $\Gamma =\gamma ({\mathbb{S}}^1)\subset {\mathbb{R}}^2$, 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 \[ \mathcal A(γ):=\inf\left\{\left. \int_{B_1(0)}|\mathrm{det}(\nabla u)| \mathrm{d} x \ \right| \ u=γ\text{ on }\mathbb{S}^1\right\}, \] 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 $\Gamma$, i.e., a solution of the Plateau problem of disk type for the oriented contour $\Gamma$. 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