Foreword

Summary

The Cauchy problem (whose name was coined by Jacques Hadamard in his classical treatise Lectures on Cauchy's Problem in Linear Partial Differential Equations published in the Silliman Lecture Series by Yale University Press in 1921) is one of the major problems of the theory of partial differential equations, both in its classical form as it arose in the late nineteenth and early twentieth centuries and in the modern theory, which has seen such a meteoric development since the Second World War. In the classical period, it appeared in two significantly different forms: first as the basic formulation for the most fundamental result in the theory of partial differential equations in the analytic domain—the Cauchy–Kowalewski theorem—as well as the classical boundary value problem, which was relevant to the study of both the wave equation and the more general class of second-order equations of hyperbolic type. In the Cauchy–Kowalewski theorem, the basic local existence theorem for a general (or in the classical case, general second-order) partial differential equation in analytic form with the highest normal derivative near a point on a surface written in terms of derivatives of lower normal-order is given in terms of the Cauchy data—that is, the prescription of the lower normal derivatives on the surface. The Cauchy problem for the wave equation is solvable globally (i.e., for the whole space, or at the very least a non-microscopic region) in terms of Cauchy data on a non-characteristic surface.