Multiple solutions to the Plateau problem for nonconstant mean curvature
A closed circle \(\Gamma\) on the unit sphere \(S^ 2\) in \(\mathbb{R}^ 3\) obviously bounds two different surfaces of constant mean curvature one. Under certain conditions on the diameter of the Jordan arc \(\Gamma\) and the size of the constant \(H\) it is true in general that there always exist two geometrically different surfaces of mean curvature \(H\) bounded by \(\Gamma\). This was proved by \textit{H. Brezis} and \textit{J.-M. Coron} [Commun. Pure Appl. Math. 37, 149-187 (1984; Zbl 0537.49022)], \textit{M. Struwe} [Arch. Ration. Mech. Anal. 93, 135-157 (1986; Zbl 0603.49027)], and \textit{K. Steffen} [Arch. Ration. Mech. Anal. 94, 101-122 (1986; Zbl 0678.49036)] about ten years ago. A generalization to the case of nonconstant mean curvature was given by \textit{M. Struwe} [Analysis, et cetera, Res. Pap. in Honor of J. Moser's 60th Birthd., 639-666 (1990; Zbl 0703.53049)]. In this paper the authors improve Struwe's result in the following sense: If \(H_ 0\) is a constant such that there exist two solutions then there is a number \(\alpha > 0\) such that for any curvature function \(H \in C^ 1(\mathbb{R}^ 3,\mathbb{R})\) which satisfies \(\| H - H_ 0\|_{L^ \infty(\mathbb{R}^ 3,\mathbb{R})} < \alpha\) there exist at least two solutions. They treat the Dirichlet problem as well as the Plateau problem. The proof uses the Mountain-Pass-Lemma and requires a careful analysis of the behaviour of Palais-Smale-Sequences. The paper also contains a short section on some applications to other geometrical problems. Similar results have independently been obtained by \textit{N. Jakobowsky} in [Math. Z. 217, 497-512 (1994; Zbl 0814.53008)] and [J. Reine Angew. Math. 457, 1-21 (1994; Zbl 0809.53066)].
scientific article; zbMATH DE number 572046
scientific article