Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the well-posedness of a $ C\sp \infty$ Goursat problem for a partial differential operator of order greater than two


Author: Jaime Carvalho e Silva
Journal: Proc. Amer. Math. Soc. 94 (1985), 612-616
MSC: Primary 35E15
DOI: https://doi.org/10.1090/S0002-9939-1985-0792271-9
MathSciNet review: 792271
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form

$\displaystyle {C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y})$

to be $ {C^\infty }$-well posed, showing at the same time that a necessary and sufficient condition of Hasegawa cannot be extended. The result can be generalised to operators of higher orders but leads to cumbersome conditions; nevertheless, we show that the condition of Hasegawa is also not sufficient in this case.

References [Enhancements On Off] (What's this?)

  • [1] Yukiko Hasegawa, On the 𝐶^{∞}-problem for equations with constant coefficients, J. Math. Kyoto Univ. 19 (1979), no. 1, 125–151. MR 527399
  • [2] L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin, 1963.
  • [3] Tatsuo Nishitani, On the \cal𝐸-well posedness for the Goursat problem with constant coefficients, J. Math. Kyoto Univ. 20 (1980), no. 1, 179–190. MR 564677
  • [4] S. Leif Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat. 8 (1969), 145–162. MR 0271538, https://doi.org/10.1007/BF02589555

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35E15

Retrieve articles in all journals with MSC: 35E15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792271-9
Article copyright: © Copyright 1985 American Mathematical Society