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Hypoellipticity for a class of the second order partial differential equations

Published online by Cambridge University Press:  22 January 2016

Tadato Matsuzawa
Tadato Matsuzawa*
Affiliation:
Mathematical Institute, Nagoya University
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In this paper, we shall investigate the hypoellipticity for a class of degenerate equations of the second order with complex coefficients as a direct extension of the results obtained in [8]. As is well known, the satisfactory general results about hypoellipticity of real operators of the second order have been obtained in [3] and [9], where the assumption that the operators are real plays a crucial role and our aim of this paper is to study the operators with complex coefficients. Our method may be considered as a generalization of the usual variational method replacing the Gårding inequality by the estimate (2.15), (cf. [3], [5]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 66 , May 1977 , pp. 1 - 12
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

References

[1] Freidlin, M. I.: On the factorization of nonnegative definite matrices, Theor. Probability Appl., 13 (1968), 354–356.Google Scholar
[2] Hörmander, L.: Linear partial differential operators, Springer Verlag, 1964.Google Scholar
[3] Hörmander, L.: Hypoelliptic second order differential equations, Acta Math., 119 (1968), 147–171.Google Scholar
[4] Hörmander, L.: Fourier integral operators, I, Acta Math., 127 (1971), 79–183.Google Scholar
[5] Kato, Y.: On a class of hypoelliptic differential operators, Proc. Japan Acad., 46, No. 1 (1970), 33–37.Google Scholar
[6] Kohn, J. J.: Pseudo-differential operators and hypoellipticity, Berkeley Symposium (1972).Google Scholar
[7] Matsuzawa, T.: On some degenerate parabolic equations I, Nagoya Math. J. 51 (1973), 57–77, II, Nagoya Math. J. 52 (1973), 61–84.CrossRefGoogle Scholar
[8] Matsuzawa, T.: On a hypoelliptic boundary value problem, Nagoya Math. J. (to appear).Google Scholar
[9] Oleinkik, O. A. and Radkevic, E. V.: Second order equations with nonnegative characteristic form, Amer. Math. Soc, 1973.Google Scholar