Compactness and gradient bounds for solutions of the mean curvature system in two independent variables
Let \(\Omega\) denote an open set in the plane and assume that \(n \geq 4\). Let \(M \subset \mathbb{R}^ n\) be the graph of the \(C^ 2\)-mapping \(u : \Omega \to \mathbb{R}^{n-2}\), i.e. \(M = \{(x,u(x)) \in \mathbb{R}^ n : x \in \Omega\}\). As a first result the author shows \( (D_ r(x_ 0) \subset \Omega)\) \[ \sup_{x \in D_{r/2}} \biggl( \bigl | Du(x) \bigr | + \bigl | {\mathcal K} (x) \bigr | \biggr) \leq C(r,{\underset D_ {r} {\text{osc}}} u) \] for solutions \(u\) of the minimal surface system where \({\mathcal K}\) is the Gaussian curvature. In further sections the compactness properties of \(C^ 2\)-solutions of the mean curvature system (with \(H \not \equiv 0)\) are investigated: For example, assume that \(u_ l : \Omega \to \mathbb{R}^{n-2}\) are solutions of the mean curvature system such that the areas are uniformly bounded and \(| H_ l (x) | \cdot (1 + | Du_ l (x) |^ 2)^{1/2} \leq \text{const}\). Then there is an integer multiplicity varifold \(M\) such that \(M_ l \to M\). Apart from a descrete set of points \(M\) is a smooth surface and the convergence is uniform. This result is in turn obtained with the help of a suitable a priori estimate for the second fundamental form. A final section contains a careful analysis of the possible singular set.
scientific article; zbMATH DE number 653342
scientific article