On the singular planar Plateau problem

Abstract

Given any $Γ=γ(\mathbb{S}^1)\subset\mathbb{R}^2$, image of a Lipschitz curve $γ:\mathbb{S}^1\rightarrow \mathbb{R}^2$, 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 $u:B_1(0)\rightarrow \mathbb{R}^2$ 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

Figures (7)

• Figure 1: A non-injective curve with a support set $\Gamma$, such that $\mathbb{R}^2\setminus \Gamma$ consists of three bounded connected components. In terms of equivalence classes in $\pi_1$, we have $\gamma\equiv \sigma_1\sigma_2^{-1}\sigma_3$ in $\pi_1(\mathbb{R}^2\setminus{P_1,P_2,P_3})$.
• Figure 2: A non-injective curve whose support $\Gamma$ is such that $\mathbb{R}^2\setminus \Gamma$ has $3$ bounded connected components. As equivalence class we have $\gamma\equiv \sigma_1\sigma_2\sigma_3\sigma_2^{-1}$ in $\pi_1(\mathbb{R}^2\setminus\{P_1,P_2,P_3\})$ and thus $(1,0,1)\in \mathrm{Ad}(\gamma)$.
• Figure 3: A non-injective curve supported on a $\Gamma\subset \mathbb{R}^2\setminus\{P_1,P_2\}$ with a representing word $\gamma\equiv \sigma_1\sigma_2\sigma_1^{-1}\sigma_2^{-1}$. Here both $(0,2), (2,0)\in\mathrm{Ad}(\gamma)$ but the optimum is given by $(0,2)$ since $|U_1|>|U_2|$.
• Figure 4: A depiction of the situation in the proof of Lemma \ref{['teo_2.18']}. The main issue is to link $\alpha$, $\beta$ to the same base point $P$. To do that, the homotopy $\Phi_{\alpha,\beta}$ is exploited to produce the dotted curve $\Phi_{\alpha,\beta}(1-\cdot,(1,0))$, which is then concatenated to $\hat{\alpha}$ to obtain $\hat{\gamma}$ (in red).
• Figure 5: A depiction of the construction of $\eta$ in Corollary \ref{['cor2.35']} in the case $n=2$: the function $\Psi$ identifies a curve $\Psi(\cdot,\theta):[0,1]\rightarrow\mathbb{R}^2\setminus \{P_1,P_2\}$ (the dotted line in the figure) connecting $\nu$ (the solid dark line) to $\zeta$ (the solid grey line) with starting and ending points $\Psi(0,\theta)=\nu(\theta)$, $\Psi(1,\theta)=\zeta(\theta)$. The curve $\eta$ is then the suitable concatenation of $\nu$, $\zeta$ and $\Psi(\cdot,\theta)$ and their inverses.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof
• Corollary 2.6