A Walk Through Partial Differential Equations

Abstract

This is an informal talk for non-specialists dealing on an elementary level with some aspects of the theory of partial differential equations. In recent years progress in the theory has been tremendous, often in unexpected directions, while also solving classical problems in more general settings. New fields have been added, like the study of variational inequalities, of solitons, of wave front sets, of pseudo-differential operators, of differential forms on manifolds, etc. Much of the progress has been made possible by the use of functional analysis. However, in the process much of the original simplicity of the theory has been lost. This is perhaps connected with the emphasis on solving problems, which often requires the piling up mountains of a priori inequalities and the skillful juggling of function spaces to make ends meet. It is good to remember that mathematics is not only concerned with solving problems, but with studying the structure and behavior of the objects it creates. One of the best examples is the classical theory of functions of a complex variable. It, incidentally, does solve problems as in the Riemann mapping theorem. But much of its beauty lies in statements that can hardly be considered as “solving” anything, like the calculus of residues, or Picard’s theorem, or Cauchy’s formula