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 Mat. Sb. (N.S.), 1973, Volume 91(133), Number 3(7), Pages 367–389 (Mi msb3301)

On a class of globally hypoelliptic operators

A. V. Fursikov

Abstract: We consider an operator $A$ which is defined on an $(n+1)$-dimensional manifold $\Omega$ and which is elliptic everywhere outside an $n$-dimensional submanifold $\Gamma$. If $(x)$ represents the local coordinates in $\Gamma$ and $t$ is the distance to $\Gamma$, then in the coordinates $(x,t)$ the operator $A$ is of the form
$$Au=\sum_{|\beta|+l\leqslant m}a_{\beta l}(x,t)t^{lq}D^\beta_xD^l_tu,$$
where $q>1$ is an integer. We present a necessary and sufficient condition for infinite differentiability in a neighborhood of $\Gamma$ of the solution of $Au=f$ if $f$ is infinitely differentiable in a neighborhood of $\Gamma$.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1973, 20:3, 383–405

Bibliographic databases:

UDC: 517.944
MSC: Primary 58G99, 35B99, 35H05; Secondary 58G15

Citation: A. V. Fursikov, “On a class of globally hypoelliptic operators”, Mat. Sb. (N.S.), 91(133):3(7) (1973), 367–389; Math. USSR-Sb., 20:3 (1973), 383–405

Citation in format AMSBIB
\Bibitem{Fur73} \by A.~V.~Fursikov \paper On~a~class of globally hypoelliptic operators \jour Mat. Sb. (N.S.) \yr 1973 \vol 91(133) \issue 3(7) \pages 367--389 \mathnet{http://mi.mathnet.ru/msb3301} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=330743} \zmath{https://zbmath.org/?q=an:0299.35020} \transl \jour Math. USSR-Sb. \yr 1973 \vol 20 \issue 3 \pages 383--405 \crossref{https://doi.org/10.1070/SM1973v020n03ABEH001881}