Cauchy-Kovalevsky Theorem

Theorem

Let $\KK$ denote the field of either the real or complex numbers.

Let $V = \KK^m$.

Let $W = \KK^n$.

Let $A_1, A_2, \ldots, A_{n − 1}$ be analytic functions defined on some neighborhood of $\tuple {0, 0}$ in $W \times V$, taking values in the $m \times m$ matrices.

Let $b$ be an analytic function with values in $V$ defined on the same neighborhood.

Then there exists a neighborhood of $0$ in $W$ on which the quasilinear Cauchy problem:

$\partial_{x_n} f = \map {A_1} {x, f} \partial_{x_1} f + \cdots + \map {A_{n - 1} } {x, f} \partial_{x_{n - 1} } f + \map b {x, f}$

with initial condition:

$\map f x = 0$

on the hypersurface:

$x_n = 0$

has a unique analytic solution:

$f: W \to V$

near $0$.

Proof

This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.