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 Mat. Sb. (N.S.), 1983, Volume 121(163), Number 4(8), Pages 562–575 (Mi msb2225)

Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator

A. I. Karol'

Abstract: In this paper the author constructs an asymptotic expansion of the resolvent of the operator of the Dirichlet problem for an elliptic equation of divergence form with a power degeneracy on the boundary. To construct the expansion a variant of the technique of pseudodifferential operators ($\Psi$DO's) with operator-valued symbols is used, in combination with the technique of “ordinary” scalar $\Psi$DO's. The difference between the resolvent and the approximation thus obtained is an integral operator whose kernel decreases at infinity faster than any power of the spectral parameter. In a neighborhood of the boundary this operator smooths only in directions tangent to the boundary.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 49:2, 553–567

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J70; Secondary 35P20

Citation: A. I. Karol', “Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator”, Mat. Sb. (N.S.), 121(163):4(8) (1983), 562–575; Math. USSR-Sb., 49:2 (1984), 553–567

Citation in format AMSBIB
\Bibitem{Kar83} \by A.~I.~Karol' \paper Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator \jour Mat. Sb. (N.S.) \yr 1983 \vol 121(163) \issue 4(8) \pages 562--575 \mathnet{http://mi.mathnet.ru/msb2225} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=716114} \zmath{https://zbmath.org/?q=an:0553.35036} \transl \jour Math. USSR-Sb. \yr 1984 \vol 49 \issue 2 \pages 553--567 \crossref{https://doi.org/10.1070/SM1984v049n02ABEH002727} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. I. Karol', “The zeta-function of a degenerate elliptic operator”, Math. USSR-Sb., 52:1 (1985), 209–230
2. Solomiak M., “The Spectral Asymptotics of the Schrodinger Operator with Irregular Homogeneous Potential”, 278, no. 2, 1984, 291–295
3. Levendorskii S., “The Approximate Spectral Projection Method”, Acta Appl. Math., 7:2 (1986), 137–197
4. S. Z. Levendorskii, “Non-classical spectral asymptotics”, Russian Math. Surveys, 43:1 (1988), 149–192
5. Evgueniya Dyachenko, Nikolai Tarkhanov, “Degeneration of Boundary Layer at Singular Points”, Zhurn. SFU. Ser. Matem. i fiz., 6:3 (2013), 283–297
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