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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonanalytic solutions of certain linear PDEs


Author: E. C. Zachmanoglou
Journal: Trans. Amer. Math. Soc. 277 (1983), 805-814
MSC: Primary 35A07; Secondary 35B65
MathSciNet review: 694389
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Abstract: It is shown that if $ P$ is a linear partial differential operator with analytic coefficients, and if $ M$ is an analytic submanifold of codimensions $ 3$ in $ {{\mathbf{R}}^n}$, which is partially characteristic with respect to $ P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $ M$, solutions of the equation $ Pu = 0$ which are flat or singular precisely on $ M$. The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of $ P$.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0694389-0
Article copyright: © Copyright 1983 American Mathematical Society