Compactness and gradient bounds for solutions of the mean curvature system in two independent variables
type: Item
instance of: scholarly article
title: Compactness and gradient bounds for solutions of the mean curvature system in two independent variables
zbMATH Open document ID: 0813.35010
DOI: 10.1007/BF02921585
author: Giovanni Gregori
published in: The Journal of Geometric Analysis
publication date: 1994-11-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 653342
zbMATH Keywords: gradient estimates
zbMATH Keywords: minimal surface system
zbMATH Keywords: mean curvature system
reviewed by: M. Fuchs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 653342