Cauchy problem

For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.