Sternin, B. Yu.; Shatalov, V. E. 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 [Tr. Mosk. Mat. O.-va 15, 346–382 (1966; Zbl 0161.08504); translation in Trans. Mosc. Math. Soc. 15, 387-429 (1966); Sov. Math., Dokl. 5 (1964), 1658–1661 (1965; Zbl 0138.36004); translation from Dokl. Akad. Nauk SSSR 159, 992–994 (1964); Sov. Math., Dokl. 8, 41–45 (1967; Zbl 0177.37103); translation from Dokl. Akad. Nauk SSSR 172, 44–47 (1967); Sov. Math., Dokl. 17(1976), 1306–1309 (1977; Zbl 0365.58017); translation from Dokl. Akad. Nauk SSSR 230, 287–290 (1976)]. 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. Reviewer: Vladimir N. Karpushkin (Moskva) Cited in 3 ReviewsCited in 8 Documents 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 Keywords:smooth embedding; relative ellipticity; theorem of finiteness; index formula; Sobolev problems Citations:Zbl 0161.08504; Zbl 0162.15601; Zbl 0138.36004; Zbl 0177.37103; Zbl 0365.58017 PDFBibTeX XMLCite \textit{B. Yu. Sternin} and \textit{V. E. Shatalov}, Sb. Math. 187, No. 11, 1691--1720 (1996; Zbl 0882.58053); translation from Mat. Sb. 187, No. 11, 115--144 (1996) Full Text: DOI