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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Solutions of partial differential equations with support on leaves of associated foliations
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by E. C. Zachmanoglou PDF
Trans. Amer. Math. Soc. 180 (1973), 415-421 Request permission

Abstract:

Suppose that the linear partial differential operator $P(x,D)$ has analytic coefficients and that it can be written in the form $P(x,D) = R(x,D)S(x,D)$ where $S(x,D)$ is a polynomial in the homogeneous first order operators ${A_1}(x,D), \cdots ,{A_r}(x,D)$. Then in a neighborhood of any point ${x^0}$ at which the principal part of $S(x,D)$ does not vanish identically, there is a solution of $P(x,D)u = 0$ with support the leaf through ${x^0}$ of the foliation induced by the Lie algebra generated by ${A_1}(x,D), \cdots ,{A_r}(x,D)$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.
References
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 415-421
  • MSC: Primary 35R99; Secondary 57D30, 58G99
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0320565-6
  • MathSciNet review: 0320565