Conference abstracts

Session B3 - Symbolic Analysis

July 14, 18:00 ~ 18:25

A generalization of an integrability theorem of Darboux

Irina Kogan

North Carolina State University, USA   -

In Chapter I, Livre III of his monograph "Systèmes Orthogonaux'' Darboux stated three integrability theorems. They provide local existence and uniqueness of solutions to systems of first order Partial Differential Equations of the type: \[\partial_{x_i} u_\alpha(x)=f^\alpha_i(x,u(x)), \quad i\in I_\alpha\subset\{1,\dots,n\}\] where $x_i$, $i=1,\dots, n$ are independent variables, and $u_\alpha$, $\alpha=1,\dots,m$ are dependent variables and $f^\alpha_i(x,u(x))$ are some given functions. For each dependent variable $u^\alpha$, the system prescribes partial derivatives in certain coordinate directions given by a subset $I_\alpha$. The data, near a given point $\bar x\in{\mathbb R}^n$, prescribe each of the unknown functions $u_\alpha$ along the affine subspace spanned by the coordinate vectors complimentary to the coordinate vector defined by indices $I_\alpha$.

Darboux's first theorem applies to determined systems, in which case $|I_\alpha|=1$ for all $\alpha$, while his second theorem is Frobenius' Theorem for complete systems, in which case $|I_\alpha|=n$ for all $\alpha$. The third theorem addresses the general situation where $I_\alpha$ are arbitrary subsets varying with $\alpha$. Under the appropriate integrability conditions, Darboux proved his third theorem in the cases $n=2$ and $n=3$. However, his argument does not appear to generalize in any easy manner to cases with more than three independent variables.

In the present work, we formulate and prove a theorem that generalizes Darboux's third theorem to systems of the form \[{\mathbf r}_i(u_\alpha)\big|_x = f_i^\alpha (x, u(x)), \quad i\in I_\alpha\subset\{1,\dots,n\}\] where $\{{\mathbf r}_i\}_{i=1}^n$ is an arbitrary local frame of vector-fields near $\bar x$. Furthermore, the data for $u^\alpha$ can be prescribed along an arbitrary submanifold through $\bar x$ transversal to the subset of vector-fields $\{{\mathbf r}_i\,|\, i\in I_\alpha\}$. Our proof applies to any number of independent variables and uses a nonstandard application of Picard iteration. The approach requires only $C^1$ smoothness of the $f_i^\alpha$ and the initial data. We note that, in the analytic case, this result can be proved using Cartan-Kähler theorem.

Joint work with Michael Benfield, San Diego, USA and Helge Kristian Jenssen, Pennsylvania State University, USA.

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FoCM 2017, based on a nodethirtythree design.