Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 497-524.

We study the geometry of multidimensional scalar 2 nd order PDEs (i.e. PDEs with n independent variables), viewed as hypersurfaces in the Lagrangian Grassmann bundle M (1) over a (2n+1)-dimensional contact manifold (M,𝒞). We develop the theory of characteristics of in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of . After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:

det 2 f x i x j - b ij x,f,f=0.

We show that any MAE of this class is associated with an n-dimensional subdistribution 𝒟 of the contact distribution 𝒞, and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method to solve Cauchy problems for this kind of equations.

Nous étudions la géométrie des équations aux dérivées partielles scalaires du deuxième ordre multidimensionnelles (c’est-à-dire, EDP avec n variables indépendantes), considérées comme hypersurfaces dans le fibré Grassmannien Lagrangien M (1) sur une variété de contact (2n+1)-dimensionnelle (M,𝒞). Nous développons la théorie des caractéristiques de en termes de la géométrie de contact et de la géométrie du fibré Grassmannien Lagrangien et étudions leur relation avec les intégrales intermédiaires de . Après avoir appliqué tels résultats aux équations de Monge-Ampère générales (EMA), nous concentrons notre attention sur les EMA du type introduit par Goursat en 1899 :

det 2 f x i x j - b ij x,f,f=0.

Nous montrons que toutes les EMA de cette classe sont associées à une sous-distribution n-dimensionnelle 𝒟 de la distribution de contact 𝒞 et vice-versa. Nous caractérisons les équations du type de Goursat avec leurs intégrales intermédiaires en fonction de leurs caractéristiques et donnons un critère d’équivalence locale de contact. Enfin, nous développons une méthode pour résoudre les problèmes de Cauchy pour ce genre d’équations.

DOI: 10.5802/aif.2686
Classification: 53D10, 35A30, 58A30, 58A17
Keywords: Hypersurfaces of Lagrangian Grassmannians, contact geometry, subdistributions of a contact distribution, Monge-Ampère equations, characteristics, intermediate integrals
Mot clés : hypersurfaces du fibré Grassmannien Lagrangien, géométrie de contact, sous-distribution de la distribution de contact, équations de Monge-Ampère, caractéristiques, intégrales intermédiaires
Alekseevsky, Dmitri V. 1; Alonso-Blanco, Ricardo 2; Manno, Gianni 3; Pugliese, Fabrizio 4

1 University of Edinburgh School of Mathematics and Maxwell Institute for Mathematical Sciences The Kings Buildings, JCMB Mayfield Road Edinburgh, EH9 3JZ (UK)
2 Universidad de Salamanca Departamento de Matemáticas plaza de la Merced 1-4 37008 Salamanca (Spain)
3 Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni via Cozzi 53 20125 Milano (Italy)
4 Università di Salerno Dipartimento di Matematica via Ponte don Melillo 84084 Fisciano (Italy)
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     title = {Contact geometry of multidimensional {Monge-Amp\`ere} equations: characteristics, intermediate integrals and solutions},
     journal = {Annales de l'Institut Fourier},
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Alekseevsky, Dmitri V.; Alonso-Blanco, Ricardo; Manno, Gianni; Pugliese, Fabrizio. Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 497-524. doi : 10.5802/aif.2686. http://www.numdam.org/articles/10.5802/aif.2686/

[1] Akivis, Maks; Goldberg, Vladislav Conformal differential geometry and its generalizations, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1996 (A Wiley-Interscience Publication) | MR | Zbl

[2] Alonso-Blanco, R.J.; Manno, G.; Pugliese, F. Normal forms for Lagrangian distributions on 5-dimensional contact manifolds, Differential Geom. Appl., Volume 27 (2009) no. 2, pp. 212-229 | DOI | MR

[3] Alonso Blanco, R.J.; Manno, Gianni; Pugliese, Fabrizio Contact relative differential invariants for non generic parabolic Monge-Ampère equations, Acta Appl. Math., Volume 101 (2008) no. 1-3, pp. 5-19 | DOI | MR

[4] Boillat, Guy Le champ scalaire de Monge-Ampère, Norske Vid. Selsk. Forh. (Trondheim), Volume 41 (1968), pp. 78-81 | MR | Zbl

[5] Boillat, Guy Sur l’équation générale de Monge-Ampère à plusieurs variables, C. R. Acad. Sci. Paris Sér. I Math., Volume 313 (1991) no. 11, pp. 805-808 | MR | Zbl

[6] Doubrov, B.; Ferapontov, E. V. On the integrability of symplectic Monge-Ampère equations, J. Geom. Phys., Volume 60 (2010) no. 10, pp. 1604-1616 | DOI | MR

[7] Ferapontov, Evgeny Vladimirovich; Hadjikos, Lenos; Khusnutdinova, Karima Robertovna Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, Int. Math. Res. Not. IMRN (2010) no. 3, pp. 496-535 | MR

[8] Forsyth, Andrew Russell Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Six volumes bound as three, Dover Publications Inc., New York, 1959 | MR | Zbl

[9] Goursat, E. Leçons sur l’intégration des équations aux dérivées partielles du second ordre, 1, Gauthier-Villars, Paris, 1890

[10] Goursat, E. Sur les équations du second ordre à n variables analogues à l’équation de Monge-Ampère, Bull. Soc. Math. France, Volume 27 (1899), pp. 1-34 | Numdam | MR

[11] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978 (Pure and Applied Mathematics) | MR | Zbl

[12] Kushner, Alexei Classification of Monge-Ampère equations, Differential equations: geometry, symmetries and integrability (Abel Symp.), Volume 5, Springer-Verlag, Berlin, 2009, pp. 223-256 | MR

[13] Kushner, Alexei; Lychagin, Valentin; Rubtsov, Vladimir Contact geometry and non-linear differential equations, Encyclopedia of Mathematics and its Applications, 101, Cambridge University Press, Cambridge, 2007 | MR

[14] Lax, P. D.; Milgram, A. N. Parabolic equations, Contributions to the theory of partial differential equations (Annals of Mathematics Studies, no. 33), Princeton University Press, Princeton, N. J., 1954, pp. 167-190 | MR | Zbl

[15] Lyčagin, V. Contact geometry and second-order nonlinear differential equations, Uspekhi Mat. Nauk, Volume 34 (1979) no. 1(205), pp. 137-165 | MR | Zbl

[16] Machida, Y.; Morimoto, T. On decomposable Monge-Ampère equations, Lobachevskii J. Math., Volume 3 (1999), p. 185-196 (electronic) Towards 100 years after Sophus Lie (Kazan, 1998) | MR | Zbl

[17] Morimoto, Tohru Monge-Ampère equations viewed from contact geometry, Symplectic singularities and geometry of gauge fields (Warsaw, 1995) (Banach Center Publ.), Volume 39, Polish Acad. Sci., Warsaw, 1997, pp. 105-121 | MR | Zbl

[18] Muñoz Díaz, J. Ecuaciones diferenciales I (1982) (Ed. Universidad de Salamanca)

[19] Petrovski, I. G. Lectures on partial differential equations (1991) (Dover Publication, New York)

[20] Ruggeri, Tommaso Su una naturale estensione a tre variabili dell’equazione di Monge-Ampère, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Volume 55 (1973), p. 445-449 (1974) | MR | Zbl

[21] Valiron, Georges The classical differential geometry of curves and surfaces, Lie Groups: History, Frontiers and Applications, Series A, XV, Math Sci Press, Brookline, MA, 1986 (Translated from the second French edition by James Glazebrook, With a preface by Robert Hermann) | MR | Zbl

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