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