On Two Theorems of Darboux

TL;DR

This paper rigorously formulates and proves two of Darboux's theorems on local existence and uniqueness of solutions for certain first-order PDE systems, extending results to non-analytic cases and arbitrary dimensions.

Contribution

It provides precise proofs of Darboux's theorems, including the overdetermined case, with explicit conditions and applicability to non-analytic situations and higher dimensions.

Findings

Proved local existence and uniqueness for determined systems using Picard iteration.

Extended Darboux's overdetermined systems theorem to any number of variables.

Connected the results to the classical Frobenius theorem as a special case.

Abstract

We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation contains a single derivative for which it is solved: \[\frac{\partial u_i}{\partial x_j}(x)=f_{ij}(x,u(x)).\] The data prescribe values for the unknowns $u_i$ along certain hyperplanes through a given point $\bar x$. The first theorem applies to determined systems (the number of equations equals the number unknowns), and a unique, local solution is obtained via Picard iteration. While Darboux's statement of the theorem leaves unspecified "certaines conditions de continuit\'e," it is clear from his proof that he assumes Lipschitz continuity of the maps $f_{ij}$. On the other hand, he did not address the regularity of the data. We provide a precise formulation and proof of his first theorem. The second theorem is more involved and applies to overdetermined systems of the same general form. Under the appropriate integrability conditions, Darboux used his first theorem to treat the cases with two and three independent variables. We provide a proof for any number of independent variables. While the systems are rather special, they do appear in applications; e.g., the second theorem contains the classical Frobenius theorem on overdetermined systems as a special case. The key aspect of the proofs is that they apply to non-analytic situations. In an analytic setup the results are covered by the general Cartan-K\"ahler theorem.