Mathematics – Differential Geometry
Scientific paper
2010-03-26
Mathematics
Differential Geometry
49 pages
Scientific paper
We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, 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 the equation $\mathcal{E}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal{E}$. After specifying the results to general Monge-Amp\`ere equations (MAEs), we focus our attention to MAEs of type introduced by Goursat, i.e. MAEs of the form $$ \det|\frac{\partial^2 f}{\partial x^i\partial x^j}-b_{ij}(x,f,\nabla f)\|=0. $$ We show that any MAE of the aforementioned class is associated with an $n$-dimensional subdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, and viceversa. We characterize this Goursat-type equations together with its intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method of solutions of a Cauchy problem, provided the existence of a suitable number of intermediate integrals.
Alekseevsky Dmitri
Alonso-Blanco Ricardo
Manno Gianni
Pugliese Fabrizio
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