Abstract:
Solvability in the class of multivalued solutions is investigated for Cauchy problems for hyperbolic Monge-Ampère equations. A characteristic uniformization is constructed on definite solutions of this problem, using which the existence and uniqueness of a maximal solution is established. It is shown that the characteristics in the different families that lie on a maximal solution and converge to a definite boundary point have infinite lengths. In this way a theory of global solvability is developed for the Cauchy problem for hyperbolic Monge-Ampère equations, which is analogous to the corresponding theory for ordinary differential equations. Using the same methods, a stable explicit difference scheme for approximating multivalued solutions can be constructed and a number of problems which are important for applications can be integrated by quadratures.
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