Geodesic
Multivalued solutions of hyperbolic Monge-Amp\`ere equations: solvability, integrability, approximation
Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 373-421

Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 23 titles.
Keywords: quasilinear equations, gradient blowup, complete solutions, difference approximation.
Mots-clés : maximal solutions 
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     author = {D. V. Tunitsky},
     title = {Multivalued solutions of hyperbolic {Monge-Amp\`ere} equations: solvability, integrability, approximation},
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D. V. Tunitsky. Multivalued solutions of hyperbolic Monge-Amp\`ere equations: solvability, integrability, approximation. Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 373-421. http://geodesic.mathdoc.fr/item/SM_2020_211_3_a2/