Electron. J. Diff. Eqns.,
Vol. 2007(2007), No. 65, pp. 137.
Local solvability of degenerate MongeAmpère equations
and applications to geometry
Marcus A. Khuri
Abstract:
We consider two natural problems arising in geometry which are
equivalent to the local solvability of specific equations of
MongeAmpère type. These are: the problem of locally
prescribed Gaussian curvature for surfaces in
,
and the local isometric embedding problem for twodimensional
Riemannian manifolds. We prove a general local existence result
for a large class of degenerate MongeAmpère 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.
Submitted February 28, 2007. Published May 9, 2007.
Math Subject Classifications: 53B20, 53A05, 35M10.
Key Words: Local solvability; MongeAmpère equations; isometric embeddings.
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Marcus A. Khuri
Department of Mathematics,
Stony Brook University
Stony Brook, NY 11794, USA
email: khuri@math.sunysb.edu

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