Separately twice differentiable functions and the equation of string oscillation (Q1952419)

Let \(f:\mathbb{R}^2\to \mathbb{R}\) be a separately twice differentiable function. The following is the main result in the paper. { Theorem.} Suppose that \(f"_{xx}(p)=f"_{yy}(p)\), for every \(p\in \mathbb{R}^2\). Then, there exist twice differentiable functions \(\varphi:\mathbb{R}\to \mathbb{R}\) and \(\psi:\mathbb{R}\to \mathbb{R}\) such that \(f(x,y)=\varphi(x-y)+\psi(x+y)\), for every \(x,y\in \mathbb{R}\). The obtained result may be viewed as a refinement of the ones due to \textit{V. K. Maslyuchenko} and \textit{V. V. Mykhaylyuk} [Zh. Mat. Fiz. Anal. Geom. 4, No. 2, 252--266 (2008; Zbl 1178.35133)].