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Proceedings of the American Mathematical Society

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A nonsolvable complex vector field with Hölder coefficients


Author: Howard Jacobowitz
Journal: Proc. Amer. Math. Soc. 116 (1992), 787-795
MSC: Primary 35A07
DOI: https://doi.org/10.1090/S0002-9939-1992-1107922-2
MathSciNet review: 1107922
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Abstract: It is known that the equation

$\displaystyle \frac{{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac{{\partial u}}{{\partial \xi }} = f(\xi ,\tau )$

is solvable in a neighborhood of the origin provided Im $ \alpha $ does not change sign and $ \alpha $ is at least Lipschitz smooth. An example is given where solvability fails although $ \alpha $ is of Hölder class $ \lambda $ for all $ 0 < \lambda < 1$. Further, the only solutions to

$\displaystyle \frac{{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac{{\partial u}}{{\partial \xi }} = 0$

are the constant functions.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1992-1107922-2
Article copyright: © Copyright 1992 American Mathematical Society

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