The general theory of partial differential equations in the XIXth – XXth centuries: the dialectics of its conceptual development

During of the 19th century the theory of partial differential equations developed in two main directions: i.e. the general theory of these equations and the theory of boundary problems regarding mathematical physics. Moreover, if the problem of the first category was to study the general solutions, that is to say the functions satisfying the equation, in which the expression included the arbitrary functions, and it was necessary to determine the degree of arbitrariness of the most general solution and, so as far as possible, the solution itself, thus the main problem of the second was to find the solution of the equation of a specific physical problem which satisfied certain initial and boundary conditions. In the famous «Encyklopädie der mathematischen Wissenschaften» each of these directions have been devoted its own review: E. von Weber wrote on the problems of the general theory (Bd. , H. 2 – 3; 1900) and A. Sommerfeld wrote on the boundary problems of mathematical physics (Bd. , H. 4; 1900). D.F. Egorov in the introduction to his thesis “Partial Differential Equations of the 2nd Order with two independent variables” (1899) distinguished between these areas, stressing “the fundamental difference in the statement of the problem: or this equation is regarded independently either as the equation of a special problem of physics”. In the last third of the XIXth century however the new view on the general theory of partial differential equation begins to develop. Thus, it becomes clear (S. Lie, H. Poincaré) that the general solution in “closed form” is possible only in exceptional cases. The various methods (including approximate) of the study of their solutions came on to the foreground. Little by little, the point of view on the general theory of partial differential equations as on the theory of boundary problems for different types of equations – elliptic, parabolic, hyperbolic equations, and equations of mixed type – was accepted. Although the studies on the general theory which was understood in the former sense were continued (E. Vessiot, E. Cartan, J. Drach etc.) these studies passed far from the mainstream of the development of mathematics. Thus, situation began to change in the last third of the XXth century, when these studies became the part of the modern theory of differentiable manifolds (H. Goldschmidt, S. Sternberg, A.M. Vinogradov). Moreover, once again the old problem of finding a general solution in “closed form” came on to the foreground. Although equations, which have such solutions, are a great rarity, they prove to be of particular importance in physics.