文摘
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 \(D(\rho )\backslash \{0\}\) 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 \(u''(r) =\frac{\lambda }{2}\log r+\beta (r)\); 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.