A Liouville theorem for solutions of degenerate Monge-Ampère equations
arXiv:1211.6183 · doi:10.1080/03605302.2013.814143
Abstract
In this paper, we give a new proof of a celebrated theorem of Jörgens which states that every classical convex solution of \[ \det\nabla^2 u (x)=1\quad {in} \mathbb{R}^2 \] has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Ampère equations. We prove that every convex generalized (or Alexandrov) solution of \[ \det \nabla^2 u(x_1,x_2)=|x_1|^α \quad {in} \mathbb{R}^2, \] where , has to be \[ u(x_1,x_2)= \frac{a}{(α+2)(α+1)}|x_1|^{2+α}+\frac{a b^2}{2}x_1^2 +bx_1x_2+ \frac{1}{2a} x_2^2+\ell(x_1,x_2) \] for some constants , and a linear function . This work is motivated by the Weyl problem with nonnegative Gauss curvature.
Submitted, 15 pages