Regularity and finiteness of solutions to the free boundary problem for minimal surfaces (Q800646)

Courant solved the problem of finding a minimal disc of a certain topological class with boundary on a given compact surface \(S\) embedded in \({\mathbb{R}}^ 3\). The topological class is determined by the requirement that the boundary curve of the surfaces be linked with a fixed, homotopically non trivial Jordan curve \(\Gamma\) in the complement of \(S\). In this paper we prove the following results: (I) If \(S\) is analytic then minimizing solutions are immersed up to the boundary. (II) If furthermore, \(\Gamma\) is contained in the unbounded component of \(S\), then the number of minimizing solutions is finite.