Solving of partial differential equations under minimal conditions

TL;DR

This paper characterizes solutions to the PDE rac{ ext{d}u}{ ext{d}x} + rac{ ext{d}u}{ ext{d}y} = 0 as functions depending on x-y, providing a minimal condition-based proof and extending to abstract spaces and higher-order equations.

Contribution

It offers a minimal condition characterization of solutions to a fundamental PDE and extends the approach to abstract spaces and higher-order equations.

Findings

Solutions are functions of x-y, i.e., (x,y)=re functions of x-y.

Provides a positive answer to R. Baire's question.

Extends the solution framework to abstract spaces and higher-order PDEs.

Abstract

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 of R.~Baire. Besides, we use this result to solving analogous partial differential equations in abstract spaces and partial differential equations of higher-order.