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Relative elliptic theory and the Sobolev problem. (English. Russian original) Zbl 0882.58053
Sb. Math. 187, No. 11, 1691-1720 (1996); translation from Mat. Sb. 187, No. 11, 115-144 (1996).
The article is devoted to the construction of the relative theory of pseudodifferential elliptic operators associated with a smooth embedding. The definition of relative ellipticity is introduced and the theorem of finiteness is proved. The index formula is established for relative elliptic operators. It is shown a connection between the theory of relative elliptic operators and Sobolev problems.
The article develops ideas, methods and results contained in the works by B. Yu. Sternin [Trudy Moscov. Mat. Obshch. 15, 346-382 (1966), English translation in Trans. Moscow Math. Soc. 1966 (1967; Zbl 0162.15601); Dokl. Akad. Nauk SSSR 159, 992-994, (1964), English translation in Sov. Math. Dokl. 5(1964), 1658-1661 (1965; Zbl 0138.36004); Dokl. Akad. Nauk SSSR 172, 44-47 (1967), English translation in Sov. Math. Dokl. 8, 41-45 (1967; Zbl 0177.37103); Dokl. Akad. Nauk SSSR 230, 287-290 (1976), English translation in Sov. Math. Dokl. 17 (1976), 1306-1309 (1977; Zbl 0365.58017)]. It represents, in particular, new opinions with respect to classical elliptic Sobolev problems as well as to relative elliptic theory from the point of view of the theory of modern differential equations.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S15 Boundary value problems for PDEs with pseudodifferential operators
35S30 Fourier integral operators applied to PDEs
58J05 Elliptic equations on manifolds, general theory
47A53 (Semi-) Fredholm operators; index theories
47G30 Pseudodifferential operators
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