Partial differential equation

“Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.”

“The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of l746 papers by Jean Le Rond d'Alembert (l7l7-l783), who sought the fundamental modes of vibration of a vibrating string.”

“If the original work of Calderón and Zygmund was related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to general hypoelliptic operators, and the more recent explosion of interest in the theory of oscillatory integrals and in problems involving curvature has much to do with hyperbolic equations.”