Radially Symmetric Solutions To The Graphic Willmore Surface Equation

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb{R}}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb{R}}^3$ must be flat. When $u$ is a smooth radial solution over a punctured disk $D(\rho )\mathrm{\backslash }\{0\}$ and is in $C^1(D(\rho ))$, we show that there exist a constant $\lambda$ and a function $\beta$ in $C^0(D(\rho ))$ such that $u^{\mathrm{\prime }\mathrm{\prime }}(r)=\frac{\lambda }{2}\log r+\beta (r)$; moreover, the graph of $u$ is contained in a graphical region of an inverted catenoid which is uniquely determined by $\lambda$ and $\beta (0)$. It is also shown that a radial solution on the punctured disk extends to a $C^1$ function on the disk when the mean curvature is square integrable.