Multiple solutions for the nonparametric Plateau problem within the Euclidean space $$\mathbb R^p$$ R p of arbitrary dimension

Abstract

With the aid of Dirichlet’s principle, we can minimize the area-functional for graphs in \(\mathbb R^p\) defined on convex domains \(\varOmega \subset \mathbb R^2\); here the dimension \(p\ge 3\) is arbitrary. In the minimizing sequence we replace the graphs with embeddings of the unit disc via the uniformization method, and then we substitute them by the harmonic extensions of their boundary values. This is possible within the class of harmonic embeddings due to a beautiful result by Kneser (Jahresber Dt Math Ver 35:123–124, 1926). We calculate the first and second variation of the area-functional with higher codimensions explicitly. In the beginning we present the quasilinear nonparametric minimal surface system under Dirichlet boundary conditions, where uniqueness does not seem to prevail for \(p\ge 4\). The constructed solutions above are stable in the sense that they possess a nonnegative second variation. However, we indicate by an example how to construct minimal graphs which are unstable. Furthermore, we prove a mountain-pass lemma: an absolute minimizer together with a further strict relative minimizer of the area-functional possess a third minimal graph on the mountain-pass. In comparison with the Morse theory for Dirichlet’s integral developed by R. Courant, M. Shiffman, G. Strohmer, A. Tromba, M. Struwe, J. Jost, and R. Jakob, we can consider critical points of the area-functional and apply the continuity theorem of Morse and Tompkins (Am J Math 63:825–838, 1941). We refrain from the construction of certain minimizing paths, however, we control the isothermal parameters along the admissible mountain-paths via the theory of nonlinear elliptic systems by E. Heinz.