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

This article is cited in 4 scientific papers (total in 4 papers)

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.

Full text: PDF file (2002 kB)
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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
Received: 14.11.1968

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  mathnet  crossref  mathscinet  zmath
    2. M. I. Vishik, V. V. Grushin, “Degenerating elliptic differential and psevdo-differential operators”, Russian Math. Surveys, 25:4 (1970), 21–50  mathnet  crossref  mathscinet  zmath
    3. A. V. Fursikov, “O globalnoi gladkosti reshenii odnogo klassa vyrozhdayuschikhsya ellipticheskikh uravnenii”, UMN, 26:5(161) (1971), 227–228  mathnet  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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