A non-parametric Plateau problem with partial free boundary (Q6621562)

The authors look for an area-minimizing surface which can be written as a graph over a bounded open convex set \(\Omega \subset \mathbb{R}^{2}\), and spanning a Jordan curve \(\Gamma _{\sigma }=\gamma \cup \sigma \subset \mathbb{R}^{2}\times \lbrack 0,+\infty )\) that is partially fixed, \(\gamma \) being fixed (Dirichlet condition) and given by a family \(\{\gamma _{i}\}_{i=1}^{n}\subset \partial \Omega \times \lbrack 0,+\infty )\) of \(n\in \mathbb{N}\) curves, each joining distinct pairs of points \( \{(p_{i},q_{i})\}_{i=1}^{n}\) of \(\partial \Omega \), whereas \(\sigma \), which represents the free boundary, is an unknown and consists of (the image of) \(n \) curves \(\sigma _{1},\ldots ,\sigma _{n}\) sitting in \(\Omega \), and joining the endpoints of \(\gamma \) in order that \(\gamma \cup \sigma \) forms a Jordan curve \(\Gamma _{\sigma }\) in \(\mathbb{R}^{3}\). Each curve \(\gamma _{i} \) is supposed to be Cartesian, i.e., it can be expressed as the graph of a given nonnegative function \(\varphi \) defined on a corresponding portion of \( \partial \Omega \). The minimization problem under consideration is: \( \inf_{(\sigma ,\psi )\in \mathcal{X}_{\varphi }}\int_{\Omega \setminus E(\sigma )}\sqrt{1+\left\vert \nabla \psi \right\vert ^{2}}dx\), where \( \mathcal{X}_{\varphi }=\{(\sigma ,\psi )\in \Sigma \times W^{1,1}(\Omega ):\psi =0\) a.e. in \(E(\sigma )\) and \(\psi =\varphi \) on \(\partial ^{D}\Omega \}\). The authors prove the existence result as follows:\ If \(\Omega \) is strictly convex, there exists a solution \((\sigma ,\psi )\in \mathcal{X} _{\varphi }\) to the minimization problem such that \(\psi \) is continuous on \( \Omega \), analytic in \(\Omega \setminus E(\sigma )\), and \(\Omega \cap \partial E(\sigma )\) consists of a family of mutually disjoint analytic curves (joining \(p_{i}\) and \(q_{j}\) in some order). Moreover, each connected component of \(E(\sigma )\) is convex. For the proof, the authors observe that the class \(\mathcal{X}_{\varphi }\) is not closed under the weak* convergence in the space \(BV\) and they introduce the weak formulation associated to the previous minimization problem and written as \(\inf_{(\sigma ,\psi )\in \mathcal{W}}\mathcal{F}(\sigma ,\psi )\), where \(\mathcal{W}=\{(\sigma ,\psi )\in \Sigma \times BV(\Omega ):\psi =0\) a.e. in \(E(\sigma )\}\), \(\Sigma =\{\sigma =(\sigma _{1},\ldots ,\sigma _{n})\in (\mathrm{Lip}([0,1];\overline{\Omega } ))^{n}\) that satisfies injectivity and other properties\(\}\), and \(\mathcal{F} \) is the functional defined by \(\mathcal{F}(\sigma ,\psi )=\int_{\Omega } \sqrt{1+\left\vert \nabla \psi \right\vert ^{2}}dx+\left\vert D^{s}\psi (\Omega )\right\vert -\left\vert E(\sigma )\right\vert +\int_{\partial \Omega }\left\vert \psi -\varphi \right\vert d\mathcal{H}^{1}\), \(D^{s}\psi \) being the singular part of the measure \(D\psi \). They introduce the sets \( \mathcal{W}_{\mathrm{conv}}=\{(\sigma ,\psi )\in \Sigma _{\mathrm{conv}}\times BV(\Omega ):\psi =0\) a.e. in \(E(\sigma )\}\), and \(\Sigma _{\mathrm{conv}}=\{\sigma =(\sigma _{1},\ldots ,\sigma _{n})\in \Sigma :\) \(E(\sigma _{i})\) is convex for all \( i=1,\ldots ,n\}\). They prove that they can reduce \(\mathcal{W}\) to \(\mathcal{ W}_{\mathrm{conv}}\): For every \((s,\zeta )\in \mathcal{W}\) there exists \((\sigma ,\psi )\in \mathcal{W}_{\mathrm{conv}}\) such that every connected component of \( E(\sigma )\) is convex, and \(\mathcal{F}(\sigma ,\psi )\leq \mathcal{F} (s,\zeta )\), that implies: \(\inf_{(s,\zeta )\in \mathcal{W}}\mathcal{F} (s,\zeta )=\inf_{(s,\zeta )\in \mathcal{W}_{\mathrm{conv}}}\mathcal{F}(s,\zeta )\). They prove the existence of minimizers of \(\mathcal{F}\) in \(\mathcal{W} _{\mathrm{conv}}\) and that every minimizer \((\sigma ,\psi )\) of \(\mathcal{F}\) in \( \mathcal{W}_{\mathrm{conv}}\) is such that every connected component of \(E(\sigma )\) is convex, introducing an appropriate convergence in \(\mathcal{W}_{\mathrm{conv}}\) and proving a compactness property in this space. They prove regularity properties of the minimizers. They compare the solutions to the classical Plateau problem in parametric form in the cases \(n=1,2\) to solutions to problems they introduced. The paper ends with some possible applications and open problems.