Cauchy Problem Definition

For a given m-order partial differential equation , the problem of finding a solution function u on \mathbb{R}^n that satisfies the boundary conditions that, for a smooth manifold S\subset\mathbb{R}^n, \textstyle u(x) = f_0(x) and \textstyle \frac{\partial^k u(x)}{\partial n^k} = f_k(x), \forall x\in S, k = 1\dots m-1, given specified functions f_k defined on, and vector n normal to, the manifold.