Geodesic
Local solvability of degenerate Monge-Ampère equations and applications to geometry
Electronic Journal of Differential Equations, Tome 2007 (2007).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in $\mathbb{R}^{3}$, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes and possesses a nonvanishing Hessian matrix at a critical point.
Classification : 53B20, 53A05, 35M10
Keywords: local solvability, Monge-Ampère equations, isometric embeddings 
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     author = {Khuri, Marcus A.},
     title = {Local solvability of degenerate {Monge-Amp\`ere} equations and applications to geometry},
     journal = {Electronic Journal of Differential Equations},
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     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a212/}
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Khuri, Marcus A. Local solvability of degenerate Monge-Ampère equations and applications to geometry. Electronic Journal of Differential Equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a212/