Some Problems in the Theory of Partial Differential Equations *

It was only in 1940 that Carleman proved the uniqueness theorem, in the class of n times differentiable functions, for solutions of the Cauchy problem for linear partial differential equations with two independent variables that has no multiple roots. Thus, a perfect clarification of the question of uniqueness for solutions of the Cauchy problem in classes of non-analytic functions remains one of the basic and, apparently, most difficult tasks in the theory of partial differential equations. The problem of non-unique extension of real solutions beyond a characteristic surface is one of the basic questions in the theory of partial differential equations; this problem has not been solved so far, even in the analytic case. A classical solution of a partial differential equation is a function having partial derivatives of any order involved in the equation; these derivatives are supposed to satisfy the equation.