Cauchy–Kowalevski Theorem

In mathematics, the Cauchy–Kowalevski theorem (also written as the Cauchy–Kovalevskaya theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sophie Kovalevskaya (1875).

This theorem is about the existence of solutions to a system of m differential equations in n dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables.
Let K denote either the fields of real or complex numbers, and let V = K^m and W = Kⁿ. Let A₁, ..., A_n−1 be analytic functions defined on some neighbourhood of (0, 0) in V × W and taking values in the m × m matrices, and let b be an analytic function with values in V defined on the same neighbourhood. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem

${\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}$

with initial condition

${\displaystyle f(x)=0}$

on the hypersurface

${\displaystyle x_{n}=0}$

has a unique analytic solution ƒ : W → V near 0.
Lewy's example shows that the theorem is not valid for all smooth functions.
The theorem can also be stated in abstract (real or complex) vector spaces. Let V and W be finite-dimensional real or complex vector spaces, with n = dim W. Let A₁, ..., A_n−1 be analytic functions with values in End (V) and b an analytic function with values in V, defined on some neighbourhood of (0, 0) in V × W. In this case, the same result holds.

Courtesy of Wikipedia

Cumberland Computer Services., LLC
205-467-4055
https://cumberlandcomputerservices.com/