PDE as a Unified Subject

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Abstract

Given that one of the goals of the conference is to address the issue of the unity of Mathematics, I feel emboldened to talk about a question which has kept bothering me all through my scientific career: Is there really a unified subject of Mathematics which one can call PDE? At first glance this seems easy: we may define PDE as the subject which is concerned with all partial differential equations. According to this view, the goal of the subject is to find a general theory of all, or very general classes of PDE’s. This “natural” definition comes dangerously close to what M. Gromov had in mind, I believe, when he warned us, during the conference, that objects, definitions or questions which look natural at first glance may in fact “be stupid”. Indeed, it is now recognized by many practitioners of the subject that the general point of view, as a goal in itself, is seriously flawed. That it ever had any credibility is due to the fact that it works quite well for linear PDE’s with constant coefficients, in which case the Fourier transform is extremely effective. It has also produced significant results for some general special classes of linear equations with variable coefficients. Its weakness is most evident in connection to nonlinear equations. The only useful general result we have is the Cauchy-Kowalevsky theorem, in the quite boring class of analytic solutions. In the more restrictive frameworks of elliptic, hyperbolic, or parabolic equations, some important local aspects of nonlinear equations can be treated with a considerable degree of generality. It is the passage from local to global properties which forces us to abandon any generality and take full advantage of the special features of the important equations.