Multiple surfaces of non-constant mean curvature
Given some closed rectifiable Jordan curve in \(\mathbb{R}^ 3\) we prove existence of at least two solutions to the Plateau problem for surfaces of prescribed mean curvature parametrized over the unit disk provided the prescribed mean curvature function is sufficiently near (in \(L^ \infty\) sense) a suitable non-vanishing constant: Applying minimax methods in \(W^{1,2}\), we derive a mountain pass lemma for smooth closed Jordan curves and regular (non-constant) mean curvature functions. Proving a Palais Smale condition, we establish the existence of two distinct solutions in the smooth case. The full result is obtained by approximation. As an essential tool in both steps we have a precompactness result in \(W^{1,2}_{\text{loc}} \overline {(B_ 1(0)} \backslash \{q_ 1, \dots, q_ N\})\) for Palais Smale sequences.
scientific article; zbMATH DE number 737177
scientific article