Geodesic
Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator
Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 553-567 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator},
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A. I. Karol'. Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator. Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 553-567. http://geodesic.mathdoc.fr/item/SM_1984_49_2_a17/

[1] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra variatsionnykh zadach na resheniyakh ellipticheskikh uravnenii”, Sib. matem. zh., 20:1 (1979), 3–22 | MR | Zbl

[2] Grushin V. V., “Gipoellipticheskie differentsialnye uravneniya i psevdodifferentsialnye operatory s operatornoznachnymi simvolami”, Matem. sb., 88 (130) (1972), 504–521 | Zbl

[3] Grushin V. V., “Postroenie parametriksa dlya vyrozhdayuschikhsya ellipticheskikh operatorov metodom dvukhmasshtabnykh asimptoticheskikh razlozhenii”, Funkts. analiz i ego pril., 11:2 (1977), 76–77 | MR | Zbl

[4] Karol A. I., “O $\zeta$-funktsii vyrozhdayuscheisya ellipticheskoi kraevoi zadachi Dirikhle”, Dokl. AN SSSR, 260:1 (1981), 20–22 | MR | Zbl

[5] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977, 455 pp. | MR

[6] Khardi, Littlvud, Polia, Neravenstva, IL, M., 1948, 456 pp.

[7] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978, 279 pp. | MR

[8] Agmon S., “On kernels, eigenvalues and eigenfunctions of operators related to elliptic problems”, Comm. Pure Appl. Math., 18 (1965), 627–663 | DOI | MR | Zbl

[9] Bolley P., Camus J., Pham The Lai, “Noyau, résolvante et valeurs propres d'une classe d'opérateurs elliptiques et dégénérés”, Lecture Notes in Math., 660, 1978, 33–46 | MR | Zbl

[10] Boutet de Monvel, “Boundary problems for pseudodifferential equations”, Acta Math., 126:1 (1971), 11–51 | DOI | MR | Zbl

[11] Grubb G., “La résolvante d'un problème aux limites pseudo-différentiel elliptique”, C. R. Acad. Sc. Paris Série 1, 292, 1981, 625–627 | MR | Zbl

[12] Luke G., “Pseudodifferential operators on vector bundles”, J. Diff. Equat., 12 (1972), 566–589 | DOI | MR | Zbl

[13] Métivier G., “Comportement asymptotique des valeurs propres d'opérateurs elliptiques dégénérés”, Astérisque, 34–35, Soc. Math. de France, 1976, 215–249 | MR | Zbl

[14] Seeley R. T., “The resolvent of an elliptic boundary problem”, Amer. J. Math., 91:4 (1969), 889–920 | DOI | MR | Zbl