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Transactions of the American Mathematical Society

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

 
 

 

Solutions of partial differential equations with support on leaves of associated foliations


Author: E. C. Zachmanoglou
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
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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.


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Keywords: Support of solution, foliation, hypoellipticity, uniqueness in the Cauchy problem, second order degenerate elliptic operators
Article copyright: © Copyright 1973 American Mathematical Society