Asymptotic behaviour of minimal graphs over exterior domains
The author proves the following generalization of a well-known theorem by \textit{L. Bers} [Ann. Math., II. Ser. 53, 364-386 (1951; Zbl 0043.159)]. ``If u is a \(C^ 2\)-solution of the minimal surface equation over \({\mathbb{R}}^ n\sim \Omega\), \(\Omega\) bounded open, \(3\leq n\leq 7\), then Du(x) is bounded and has a limit as \(| x| \to \infty.''\) This follows from a more general theorem valid in any dimension \(n\geq 3\) concerning tangent cones of graph u at \(\infty\). To prove this general theorem one has to show that either Du(x) is bounded or all tangent cones of graph u at \(\infty\) are cylinders of the form \(C\times {\mathbb{R}}\) where \(C\subset {\mathbb{R}}^ n\) is an (n-1)-dimensional minimizing cone with \(\partial C=0\) and \(0\in \sin g C.\) Besides standard arguments from the regularity theory for codimension one minimal surfaces one has to use a special case of a general unique tangent cone result to show that sing \(C\neq \emptyset\).
scientific article; zbMATH DE number 4028258
scientific article