Solving of Partial Differential Equations under Minimal Conditions

Анотація

It is proved that a differentiable with respect to each variable function $f: \mathbb{R}^2\to \mathbb{R}$ is a solution of the equation $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$ if and only if there exists a function $\varphi: \mathbb{R}\to \mathbb{R}$ such that $f(x,y)=\varphi(x-y)$. This gives a positive answer to a question by R. Baire. Besides, this result is used to solve analogous partial differential equations in abstract spaces and partial differential equations of higher-order.

Mathematics Subject Classification: 26B05, 35A99.