Local solvability of degenerate Monge-Ampère equations and applications to geometry
Electronic journal of differential equations, Tome 2007 (2007)
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
Keywords: local solvability, Monge-Ampère equations, isometric embeddings
@article{EJDE_2007__2007__a12,
author = {Khuri, Marcus A.},
title = {Local solvability of degenerate {Monge-Amp\`ere} equations and applications to geometry},
journal = {Electronic journal of differential equations},
year = {2007},
volume = {2007},
zbl = {1135.53009},
language = {en},
url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a12/}
}
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__a12/