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Extension of solutions for Monge-Ampère equations of hyperbolic type. (English) Zbl 0852.35004

Janeczko, Stanisław (ed.) et al., Singularities and differential equations. Proceedings of a symposium, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 33, 437-447 (1996).
We consider the Cauchy problem for real Monge-Ampère equations of hyperbolic type. If the equations are not degenerate, it is well-known that the Cauchy problem has locally a smooth solution. But we cannot expect that it admits a smooth solution in the large, especially in the hyperbolic case. This means that, if we extend the smooth solution, singularities may appear.
The aim of this talk is to construct the singularities of the solutions in the case where the equations are hyperbolic. For our aim, we have to represent the solutions explicitly. To do so, we apply the characteristic method developed principally by G. Darboux and E. Goursat. We discuss the principal difference between the equation in the two-dimensional space and equations of Monge-Ampère type in higher dimensions.
For the entire collection see [Zbl 0840.00028].

MSC:

35A20 Analyticity in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
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