Real Analysis Exchange

Separately Twice Differentiable Functions and the Equation of String Oscillation

Taras Banakh and Volodymyr Mykhaylyuk

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Abstract

We prove that for every separately twice differentiable function \(f:\mathbb {R}^2\to\mathbb{R}\) with that \(f''_{xx}=f''_{yy}\) there exist twice differentiable functions \(\varphi, \psi:\mathbb R\to\mathbb{R}\) such that \(f(x,y)=\varphi(x+y) + \psi(x-y)\).

Article information

Source
Real Anal. Exchange, Volume 38, Number 1 (2012), 133-156.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1367265644

Mathematical Reviews number (MathSciNet)
MR3083202

Zentralblatt MATH identifier
1323.26017

Subjects
Primary: 26B05: Continuity and differentiation questions
Secondary: 35A99: None of the above, but in this section

Keywords
separately differentiable functions partial differential equations

Citation

Banakh, Taras; Mykhaylyuk, Volodymyr. Separately Twice Differentiable Functions and the Equation of String Oscillation. Real Anal. Exchange 38 (2012), no. 1, 133--156. https://projecteuclid.org/euclid.rae/1367265644


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