The Symmetric Minimal Surface Equation

Abstract

For positive functions $u\in C^2(\Omega )$, where $\Omega$ is an open subset of ${\mathbb{R}}^n$, the Symmetric Minimal Surface Equation (SME), is ${\sum }_{i=1}^n{D}_i\left(\frac{D_{i}u}{\sqrt{1+|Du{|}^2}}\right)=\frac{m-1}{u\sqrt{1+|Du{|}^2}}$. Geometrically, the SME expresses the fact that the ``symmetric graph'' $SG(u)$, defined by $SG(u)=\{(x,\xi )\in \Omega \times {\mathbb{R}}^m:|\xi |=u(x)\}$, is a minimal (i.e.\ zero mean curvature) hypersurface in $\Omega \times {\mathbb{R}}^m$. A function $u\in C^1(\Omega )$ is said to be a singular solution if $u^{-1}\{0\}\ne \varnothing$, and if $u={\lim }_{j\to \infty }{u}_j$, uniformly on each compact subset of $\Omega$, where each $u_j$ is a positive $C^2(\Omega )$ solution of the SME. The present paper develops are theory of singular solutions of the SME, including existence, Hölder and Lipschitz estimates for bounded solutions, and a compactness and regularity theory. We also prove that the singular set $u^{-1}\{0\}$ is codimension at most 2.