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1997 | 39 | 1 | 161-170
Tytuł artykułu

Monge-Ampère equations and surfaces with negative Gaussian curvature

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?
Słowa kluczowe
Rocznik
Tom
39
Numer
1
Strony
161-170
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Department of Mathematics, Kyoto Sangyo University Kamigamo, Kita-ku, Kyoto 603, Japan
Bibliografia
  • [1] M.-H. Amsler, Des surfaces à courbure constante négative dans l'espace à trois dimensions et de leurs singularités , Math. Ann. 130 (1955), 234-256.
  • [2] R. Courant and D. Hilbert, Method of Mathematical Physics , vol. 2, Interscience, New York, 1962.
  • 3] G. Darboux, Leçon sur la théorie générale des surfaces , tome 3, Gauthier-Villars, Paris, 1894.
  • [4] N. V. Efimov, Generation of singularities on surfaces of negative curvature , Mat. Sb. 64 (1964), 286-320.
  • [5] E. Goursat, Leçons sur l'intégration des équations aux dérivées partielles du second ordre , tome 1, Hermann, Paris, 1896.
  • [6] E. Goursat, Cours d'analyse mathématique , tome 3, Gauthier-Villars, Paris, 1927.
  • [7] J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques , Hermann, Paris, 1932.
  • [8] D. Hilbert, Über Flächen von constanter Gausscher Krümmung , Trans. Amer. Math. Soc. 2 (1901), 87-99.
  • [9] E. Holmgen, Sur les surfaces à courbure constante négative , C. R. Acad. Sci. Paris 134 (1902), 740-743.
  • [10] S. Izumiya, Geometric singularities for Hamilton-Jacobi equation , Adv. Stud. Pure Math. 22 (1993), 89-100.
  • [11] S. Izumiya, Characteristic vector fields for first order partial differential equations , preprint.
  • [12] S. Izumiya and G. T. Kossioris, Semi-local classification of geometric singuarities for Hamilton-Jacobi equations , J. Differential Equations 118 (1995), 166-193.
  • [13] M. Kossowski, Local existence of multivalued solutions to analytic symplectic Monge- Ampère equations , Indiana Univ. Math. J. 40 (1991), 123-148.
  • [14] H. Lewy, Über das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Veränderlichen , Math. Ann. 98 (1928), 179-191.
  • [15] H. Lewy, A priori limitations for solutions of Monge-Ampère equations I, II , Trans. Amer. Math. Soc. 37 (1934), 417-434; 41 (1937), 365-374.
  • [16] V. V. Lychagin, Contact geometry and non-linear second order differential equations , Russian Math. Surveys 34 (1979), 149-180.
  • [17] T. K. Milnor, Efimov's theorem about complete immersed surfaces of negative curvature , Adv. Math. 8 (1972), 474-543.
  • [18] T. Morimoto, La géométrie des équations de Monge-Ampère , C. R. Acad. Sci. Paris Sér. I Math. 289 (1979), 25-28.
  • [19] S. Nakane, Formation of singularities for Hamilton-Jacobi equations in several space variables , J. Math. Soc. Japan 43 (1991), 89-100.
  • [20] S. Nakane, Formation of shocks for a single conservation law , SIAM J. Math. Anal. 19 (1988), 1391-1408.
  • [21] A. Pliś, Characteristics of nonlinear partial differential equations , Bull. Acad. Polon. Sci. Cl. III 2 (1954), 419-422.
  • [22] M. Tsuji, Formation of singularities for Hamilton-Jacobi equation II , J. Math. Kyoto Univ. 26 (1986), 299-308.
  • [23] M. Tsuji, Prolongation of classical solutions and singularities of generalized solutions , Ann. Inst. H. Poincarè Anal. Non Linéaire 7 (1990), 505-523.
  • [24] M. Tsuji, Formation of singularities for Monge-Ampère equations , Bull. Sci. Math. 119 (1995), 433-457.
  • [25] H. Whitney, On singularities of mappings of Euclidean spaces, I , Ann. of Math. (2) 62 (1955), 374-410.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv39z1p161bwm
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