The Plateau problem in unbounded domains
This paper is concerned with a problem of isoperimetric type: minimize \[ {\mathcal F}(E)=\int_{\Omega}| D\phi_ E| +\int | \phi_ E- \phi_{\Gamma}| dH^{n-1} \] in the class \({\mathcal E}={\mathcal E}_ v=\{E\subset \Omega:H^ n(E)=V\},\) where \(\Omega \subset {\mathbb{R}}^ n\) is open, \(\partial \Omega\) is Lipschitzian, \(\Gamma\subset \partial \Omega\), and V is a fixed real number with \(0<V<H^ n(\Omega)\). The difficulty lies in the fact that \(\Omega\) is not assumed bounded nor is it assumed that \(H^ n(\Omega)<\infty\). Two additional hypotheses are introduced: (i) \(\Gamma\) is a bounded set, (ii) There exists a sphere \(B\subset \Omega\) with \(H^ n(B)=V\). The hypothesis (ii) allows the construction of comparison sets which give useful estimates because of the isoperimetric property of spheres. The author is then able to construct a bounded minimizing sequence and prove the existence of a solution to the problem.
scientific article; zbMATH DE number 3848111