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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1976, Volume 19, Issue 5, Pages 755–761 (Mi mz7796)

An example of a locally unsolvable differential equation of quasiprincipal type with a real-valued weighted principal symbol

N. A. Shananin

Moscow Institute of Electronic Engineering

Abstract: In this note we show that the equation
$$-\{(\frac1i\frac\partial{\partial x_1})^3+(\frac1i\frac\partial{\partial x_2})^2+6i(\frac1i\frac\partial{\partial x_1})(\frac1i\frac\partial{\partial x_2})x_1\}u=f$$
is locally unsolvable at the origin of the coordinate system. This equation belongs to the class that generalizes the principal type to the case of weighted derivatives. The example is interesting because the weighted principal symbol is real (in this situation, equations of principal type are solvable) but the unsolvability depends on the behavior of the lower-order terms in a neighborhood of the zeros of the weighted principal symbol.

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English version:
Mathematical Notes, 1976, 19:5, 447–451

Bibliographic databases:

UDC: 517.9

Citation: N. A. Shananin, “An example of a locally unsolvable differential equation of quasiprincipal type with a real-valued weighted principal symbol”, Mat. Zametki, 19:5 (1976), 755–761; Math. Notes, 19:5 (1976), 447–451

Citation in format AMSBIB
\Bibitem{Sha76} \by N.~A.~Shananin \paper An example of a~locally unsolvable differential equation of quasiprincipal type with a~real-valued weighted principal symbol \jour Mat. Zametki \yr 1976 \vol 19 \issue 5 \pages 755--761 \mathnet{http://mi.mathnet.ru/mz7796} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=425313} \zmath{https://zbmath.org/?q=an:0345.35006|0336.35007} \transl \jour Math. Notes \yr 1976 \vol 19 \issue 5 \pages 447--451 \crossref{https://doi.org/10.1007/BF01142569} 

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This publication is cited in the following articles:
1. V. V. Grushin, N. A. Shananin, “Some theorems on the singularities of solutions of differential equations with weighted principal symbols”, Math. USSR-Sb., 32:1 (1977), 32–44
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