Category:Cauchy-Kovalevsky Theorem
This category contains pages concerning Cauchy-Kovalevsky 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 $0$.
Source of Name
This entry was named for Augustin Louis Cauchy and Sofia Vasilyevna Kovalevskaya.
Subcategories
This category has only the following subcategory.
C
Pages in category "Cauchy-Kovalevsky Theorem"
The following 3 pages are in this category, out of 3 total.