For a partial differential equation defined on Rn+1 and a smooth manifold S ⊂ Rn+1 of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions of the differential equation with respect to the independent variables that satisfies[2]
subject to the condition, for some value ,
where are given functions defined on the surface (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.
The Cauchy–Kowalevski theorem states that If all the functions are analytic in some neighborhood of the point , and if all the functions are analytic in some neighborhood of the point , then the Cauchy problem has a unique analytic solution in some neighborhood of the point .