Proceedings of the American Mathematical Society

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



A geometric approach to an equation of J. D'Alembert

Authors: A. Pràstaro and Th. M. Rassias
Journal: Proc. Amer. Math. Soc. 123 (1995), 1597-1606
MSC: Primary 58A15; Secondary 58G99
MathSciNet review: 1232143
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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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Keywords: Geometry of partial differential equations, singular solutions of PDE's
Article copyright: © Copyright 1995 American Mathematical Society