Smooth Local Solutions to Degenerate Hyperbolic Monge-Ampère Equations

Abstract

In this paper, we prove the existence of smooth solutions near 0 of the degenerate hyperbolic Monge-Ampere equation $$\det (D^2 u) = K(x)h(x,u,Du)$$ , where $$K(0)=0$$ , $$K\le 0$$ , $$h(0,0,0)>0$$ . We also assume that, the zero set of small perturbation of $$D_n K$$ has a simple structure. For the proof, we first transform the linearized equation into a simple form by a suitable change of variables. Then we proceed to derive a priori estimates for the linearized equation, which is degenerately hyperbolic. Finally we use Nash-Moser iteration to prove the existence of local solutions.