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 Mat. Sb. (N.S.), 1969, Volume 79(121), Number 3(7), Pages 381–404 (Mi msb3594)

A class of degenerate elliptic operators

A. V. Fursikov

Abstract: In a bounded region $G\subset R^n$ we consider an operator $A$ which is elliptic inside the region and degenerate on its boundary $\Gamma$. More precisely, the operator $A$ has the following form in the local coordinate system $(x',x_n)$, in which the boundary $\Gamma$ is given by the equation $x_n=0$ and $x_n>0$ for points in the region $G$:
$$Au=\sum_{|l'|+l_n+\beta\leqslant2m}a_{l',l_n,\beta}(x',x_n)q^\beta x_n^{l_n}D_{x'}^{l'}D_{x_n}^{l_n}u$$
where $q$ is a parameter, and
$$\sum_{|l'|+l_n+\beta=2m}a_{l',l_n,\beta}(x',0)q^\beta{\xi'}^{l'}{\xi_n}^{l^n}\ne0\quadfor\quad|\xi|+|q|\ne0.$$

The operator $A$ will be proved Noetherian in certain spaces under the condition that $|q|$ is sufficiently large. In addition, some results will be obtained relating to how the smoothness of the solution of the equation $Au=f$ depends on the magnitude of the parameter.
A theorem is formulated concerning unique solvability in approperiate spaces for a class of degenerate parabolic operators.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1969, 8:3, 357–382

Bibliographic databases:

UDC: 517.43
MSC: 47F05, 35J70, 35K65

Citation: A. V. Fursikov, “A class of degenerate elliptic operators”, Mat. Sb. (N.S.), 79(121):3(7) (1969), 381–404; Math. USSR-Sb., 8:3 (1969), 357–382

Citation in format AMSBIB
\Bibitem{Fur69} \by A.~V.~Fursikov \paper A~class of degenerate elliptic operators \jour Mat. Sb. (N.S.) \yr 1969 \vol 79(121) \issue 3(7) \pages 381--404 \mathnet{http://mi.mathnet.ru/msb3594} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=254417} \zmath{https://zbmath.org/?q=an:0185.19102|0193.07001} \transl \jour Math. USSR-Sb. \yr 1969 \vol 8 \issue 3 \pages 357--382 \crossref{https://doi.org/10.1070/SM1969v008n03ABEH002042} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. I. Vishik, V. V. Grushin, “Boundary value problems for elliptic equations degenerate on the boundary of a domain”, Math. USSR-Sb., 9:4 (1969), 423–454
2. M. I. Vishik, V. V. Grushin, “Degenerating elliptic differential and psevdo-differential operators”, Russian Math. Surveys, 25:4 (1970), 21–50
3. A. V. Fursikov, “O globalnoi gladkosti reshenii odnogo klassa vyrozhdayuschikhsya ellipticheskikh uravnenii”, UMN, 26:5(161) (1971), 227–228
4. A. S. Kalashnikov, “Some problems for linear partial differential equations with constant coefficients in the entire space and for a class of degenerate equations in a halfspace”, Math. USSR-Sb., 14:2 (1971), 186–198
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