We discuss the regularity of solutions to the
two-dimensional degenerate Monge-Amp\'ere
equation $\det D^2 u=|x|^\alpha$, with $\alpha>-2$. We show that when $\alpha>0$ solutions admit only two possible behaviors near the origin, radial and non-radial, which in turn imply $C^{2, \delta}$ regularity. For $\alpha<0$ we prove that solutions admit only the radial behavior near the origin. This is joint work with P. Daskalopoulos.
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