Separately twice differentiable functions and the equation of string oscillation

TL;DR

This paper proves that separately twice differentiable solutions to the PDE $f_{xx}''=f_{yy}''$ are necessarily sums of functions of $x+y$ and $x-y$, characterizing their form.

Contribution

It establishes a precise form for solutions of a specific PDE under separate twice differentiability, extending understanding of such functions.

Findings

Solutions are of the form $(x,y)=_1(x+y)+_2(x-y)$.

Solutions are twice differentiable functions.

The result applies to solutions of the PDE $f_{xx}''=f_{yy}''$.

Abstract

We prove that for every separately twice differentiable solution $f$ of the PDE $f"_{xx}=f"_{yy}$ is of the form $f(x,y)=\phi(x+y)+\psi(x-y)$ for some twice differentiable functions $\phi, \psi$.