Abstract:
A survey is given of papers devoted to the problem of the existence of solutions to linear differential and pseudo-differential equations of principal type. The main results in this field are due to Lewy, Hцrmander, Nirenberg, Tréves and the author. We also give a new theorem of maximal generality on local solubility of equations of principal type. By way of illustration to the exposition we mention as examples: the Lewy operator; the operator arising from the solution of the problem with directional derivatives for elliptic second order equations; non-singular operators.
\Bibitem{Ego71}
\by Yu.~V.~Egorov
\paper On the solubility of differential equations with simple characteristics
\jour Russian Math. Surveys
\yr 1971
\vol 26
\issue 2
\pages 113--130
\mathnet{http://mi.mathnet.ru/eng/rm5192}
\crossref{https://doi.org/10.1070/RM1971v026n02ABEH003823}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=296481}
\zmath{https://zbmath.org/?q=an:0213.37003}
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This publication is cited in the following 20 articles:
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Guang Shan Xu, Zhong Tai Ma, “The Problem of Non-Smooth Solutions for a Type of Parabolic Complex Equations”, AMM, 40-41 (2010), 877
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Nicolas Lerner, Luc Robbiano, “Unicite de cauchy pour des operateurs de type principal par”, J Anal Math, 44:1 (1984), 32
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Yu. V. Egorov, Ts. V. Rangelov, “Ob odnom klasse psevdodifferentsialnykh
uravnenii s dvukratnymi kharakteristikami, ne imeyuschikh reshenii”, UMN, 32:1(193) (1977), 185–186
Yu. V. Egorov, “Novoe dokazatelstvo teoremy o lokalnoi razreshimosti
uravnenii s prostymi veschestvennymi kharakteristikami”, UMN, 32:2(194) (1977), 211–212
N. A. Shananin, “K neobkhodimym usloviyam lokalnoi razreshimosti
uravnenii kvaziglavnogo tipa”, UMN, 32:2(194) (1977), 235–236
G. M. Henkin, “The Lewy equation and analysis on pseudoconvex manifolds”, Russian Math. Surveys, 32:3 (1977), 59–130
A. M. Vinogradov, B. A. Kupershmidt, “The structures of Hamiltonian mechanics”, Russian Math. Surveys, 32:4 (1977), 177–243
P. R. Popivanov, “Local solvability of pseudodifferential operators with characteristics of second multiplicity”, Math. USSR-Sb., 29:2 (1976), 193–216
N. A. Shananin, “On local solvability of equations of quasi-principal type”, Math. USSR-Sb., 26:4 (1975), 458–470