Existence of \(C^{\infty}\) local solutions for the Monge-Ampère equation. (English) Zbl 0648.35016

The authors prove the existence of \(C^{\infty}\) convex solutions to the Monge-Ampère equation \(\det (u_{ij})=f(y,u,\nabla u)\) in a neighbourhood of \(y^ 0\) in \({\mathbb{R}}^ n,\) assuming that \(f\geq 0\) near \((y^ 0,u^ 0,p^ 0)\) and that one of the following conditions is satisfied:
(I) There exists k such that \((\partial^{\alpha}f)(y^ 0,u^ 0,p^ 0)=0\) if \(| \alpha | <k\), but \((\partial^{\beta}_ yf)(y^ 0,u^ 0,p^ 0)\neq 0\) for some \(\beta\) with \(| \beta | =k.\)
(II) \(f(y,u,p)=K(y)g(y,u,p)\) with K(y \(0)=0\), \(K\geq 0\), \(g>0\), and \(K^{-1}(0)\) is contained, near \(y^ 0\), in the union of a finite number of \(C^ 1\) hypersurfaces.
The authors use a Nash-Moser method together with Bony paradifferential calculus.
Reviewer: P.Godin


35G20 Nonlinear higher-order PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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