On the well-posedness of a 𝐶^{∞} Goursat problem for a partial differential operator of order greater than two

Carvalho e Silva, Jaime

doi:10.1090/S0002-9939-1985-0792271-9

On the well-posedness of a $C^ \infty$ Goursat problem for a partial differential operator of order greater than two
HTML articles powered by AMS MathViewer

by Jaime Carvalho e Silva

Proc. Amer. Math. Soc. 94 (1985), 612-616

DOI: https://doi.org/10.1090/S0002-9939-1985-0792271-9

PDF | Request permission

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 \[ {C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y})\] 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

Yukiko Hasegawa, On the $C^{\infty }$-problem for equations with constant coefficients, J. Math. Kyoto Univ. 19 (1979), no. 1, 125–151. MR 527399, DOI 10.1215/kjm/1250522472

Linear partial differential operators

Tatsuo Nishitani, On the ${\cal E}$-well posedness for the Goursat problem with constant coefficients, J. Math. Kyoto Univ. 20 (1980), no. 1, 179–190. MR 564677, DOI 10.1215/kjm/1250522329
S. Leif Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat. 8 (1969), 145–162. MR 271538, DOI 10.1007/BF02589555