Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Alekseevsky, Dmitri V.; Alonso-Blanco, Ricardo; Manno, Gianni; Pugliese, Fabrizio

Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 497-524.

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 $ℰ$ dans le fibré Grassmannien Lagrangien $M^{\left(1\right)}$ sur une variété de contact $(2n+1)$-dimensionnelle $(M,𝒞)$. Nous développons la théorie des caractéristiques de $ℰ$ en termes de la géométrie de contact et de la géométrie du fibré Grassmannien Lagrangien et étudions leur relation avec les intégrales intermédiaires de $ℰ$. 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 :

$$det∥\frac{{\partial }^{2}f}{\partial x^i\partial x^j}-b_{ij}\left(x,f,\nabla f\right)∥=0.$$

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.

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

$$det∥\frac{{\partial }^{2}f}{\partial x^i\partial x^j}-b_{ij}\left(x,f,\nabla f\right)∥=0.$$

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.

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

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