The Symmetric Minimal Surface Equation

The symmetric minimal surface equation (SME) on a domain $\mathrm{\Omega }\subset R^n$ is the equation $\mathcal{M}u=\frac{m-1}{u\sqrt{1+|Du{|}^2}}$, where $m$ is an integer $\ge 2$ and $\mathcal{M}(u)$ is the mean curvature operator. Geometrically this equation expresses the fact that the symmetric graph $S(u)=\{(x,\xi )\in \mathrm{\Omega }\times R^m:|\xi |=u(x)\}$ is a minimal (i.e.\ zero mean curvature) hypersurface in $R^{n+m}$.

For $n\ge 2$ the SME admits singular solutions (solutions which vanish at some points but which are nevertheless locally the uniform limit of positive smooth solutions), and such singular solutions have symmetric graphs which are singular minimal hypersurfaces.

The talk will develop the theory of such singular solutions; both regularity theory (including gradient bounds) and existence theory will be discussed.