Solving of Partial Differential Equations under Minimal Conditions

Abstract

It is proved that a differentiable with respect to each variable function f : R2 → R is a solution of the equation ∂u/∂x + ∂u/∂y = 0 if and only if there exists a function φ : R → R such that f(x, y) = φ(x - y). This gives a positive answer to a question by R. Baire. Besides, this result is used to solve analogous partial di erential equations in abstract spaces and partial differential equations of higher-order.