# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A geometric approach to an equation of J. D’AlembertHTML articles powered by AMS MathViewer

by A. Pràstaro and Th. M. Rassias
Proc. Amer. Math. Soc. 123 (1995), 1597-1606 Request permission

## Abstract:

By using a geometric framework of PDE’s we prove that the set of solutions of the D’Alembert equation $( \ast )(\frac {{{\partial ^2}\log f}}{{\partial x\partial y}}) = 0$ is larger than the set of smooth functions of two variables $f(x,y)$ of the form $( \ast \ast )f(x,y) = h(x) \bullet g(y)$. This agrees with a previous counterexample by Th. M. Rassias given to a statement by C. M. Stéphanos. More precisely, we have the following result. Theorem. The set of 2-dimensional integral manifolds of PDE $( \ast )$ properly contains the ones representable by graphs of 2-jet-derivatives of functions $f(x,y)$ expressed in the form $( \ast \ast )$. A generalization of this result to functions of more than two variables is sketched also by considering the equation $(\frac {{{\partial ^n}\log f}}{{\partial {x_1} \cdots \partial {x_n}}}) = 0$.
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