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
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Also known as
The Cauchy-Kovalevsky Theorem is also known as the Cauchy-Kovalevskaya Theorem.
Source of Name
This entry was named for Augustin Louis Cauchy and Sofia Vasilyevna Kovalevskaya.
Historical Note
The Cauchy-Kovalevsky theorem was a generalization of a theorem of Augustin Louis Cauchy's on partial differential equations.
It was given by Sofia Vasilyevna Kovalevskaya in $1975$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Kovalevsky, Sonya (1850-91)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Kovalevsky, Sonya (1850-91)