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

Dmitri V. Alekseevsky[1]; Ricardo Alonso-Blanco[2]; Gianni Manno[3]; Fabrizio Pugliese[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)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 2, page 497-524
  • ISSN: 0373-0956

Abstract

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We study the geometry of multidimensional scalar 2 n d order PDEs (i.e. PDEs with n independent variables), viewed as hypersurfaces ℰ in the Lagrangian Grassmann bundle M ( 1 ) over a ( 2 n + 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 ∂ 2 f ∂ x i ∂ x j - b i j 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.

How to cite

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Alekseevsky, Dmitri V., et al. "Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions." Annales de l’institut Fourier 62.2 (2012): 497-524. <http://eudml.org/doc/251116>.

@article{Alekseevsky2012,
abstract = {We study the geometry of multidimensional scalar $2^\{nd\}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $\mathcal\{E\}$ in the Lagrangian Grassmann bundle $M^\{(1)\}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal\{C\})$. We develop the theory of characteristics of $\mathcal\{E\}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal\{E\}$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:\[ \det \Bigl \Vert \frac\{\partial ^\{2\} f\}\{\partial x^\{i\}\partial x^\{j\}\}-b\_\{ij\}\left( x,f,\nabla f\right) \Bigr \Vert =0. \]We show that any MAE of this class is associated with an $n$-dimensional subdistribution $\mathcal\{D\}$ of the contact distribution $\mathcal\{C\}$, 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.},
affiliation = {University of Edinburgh School of Mathematics and Maxwell Institute for Mathematical Sciences The Kings Buildings, JCMB Mayfield Road Edinburgh, EH9 3JZ (UK); Universidad de Salamanca Departamento de Matemáticas plaza de la Merced 1-4 37008 Salamanca (Spain); Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni via Cozzi 53 20125 Milano (Italy); Università di Salerno Dipartimento di Matematica via Ponte don Melillo 84084 Fisciano (Italy)},
author = {Alekseevsky, Dmitri V., Alonso-Blanco, Ricardo, Manno, Gianni, Pugliese, Fabrizio},
journal = {Annales de l’institut Fourier},
keywords = {Hypersurfaces of Lagrangian Grassmannians; contact geometry; subdistributions of a contact distribution; Monge-Ampère equations; characteristics; intermediate integrals; hypersurfaces of Lagrangian Grassmannians; subdistribution of a contact distribution},
language = {eng},
number = {2},
pages = {497-524},
publisher = {Association des Annales de l’institut Fourier},
title = {Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions},
url = {http://eudml.org/doc/251116},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Alekseevsky, Dmitri V.
AU - Alonso-Blanco, Ricardo
AU - Manno, Gianni
AU - Pugliese, Fabrizio
TI - Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 497
EP - 524
AB - We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of characteristics of $\mathcal{E}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal{E}$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:\[ \det \Bigl \Vert \frac{\partial ^{2} f}{\partial x^{i}\partial x^{j}}-b_{ij}\left( x,f,\nabla f\right) \Bigr \Vert =0. \]We show that any MAE of this class is associated with an $n$-dimensional subdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, 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.
LA - eng
KW - Hypersurfaces of Lagrangian Grassmannians; contact geometry; subdistributions of a contact distribution; Monge-Ampère equations; characteristics; intermediate integrals; hypersurfaces of Lagrangian Grassmannians; subdistribution of a contact distribution
UR - http://eudml.org/doc/251116
ER -

References

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