Index of Sobolev problems on manifolds with many-dimensional singularities.

*(English. Russian original)*Zbl 1298.58015
Differ. Equ. 50, No. 2, 232-245 (2014); translation from Differ. Uravn. 50, No. 2, 229-241 (2014).

The theory of Sobolev problems is a theory of linear partial differential equations having boundary conditions on submanifolds of the initial smooth closed manifold. Later on, Sobolev problems were studied by the second author when the submanifold has singularities. The aim of that paper is to derive an index formula for the problem mentioned in the title. The authors reduce the Sobolev problem to a submanifold via pseudodifferential operators (\(\psi\) do) and translators which are not \(\psi\) do. Having in mind the theory of the translators including the corresponding index formulas see [the authors, ibid. 48, No. 12, 1577–1585 (2012); translation from Differ. Uravn. 48, No. 12, 1612–1620 (2012; Zbl 1267.35269) and ibid. 49, No. 4, 494–509 (2013); translation from Differ. Uravn. 49, No. 4, 513–527 (2013; Zbl 1274.58006)], they obtain in Theorem 2 explicit index formula for the elliptic Sobolev problem under investigation.

Reviewer: Petar Popivanov (Sofia)

##### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |

58J05 | Elliptic equations on manifolds, general theory |

58J32 | Boundary value problems on manifolds |

##### Keywords:

Sobolev (co)boundary problem; index problem; elliptic Sobolev problem; manifold with singularities
PDF
BibTeX
XML
Cite

\textit{A. Yu. Savin} and \textit{B. Yu. Sternin}, Differ. Equ. 50, No. 2, 232--245 (2014; Zbl 1298.58015); translation from Differ. Uravn. 50, No. 2, 229--241 (2014)

Full Text:
DOI

##### References:

[1] | Savin, AYu; Sternin, BYu, Elliptic translators on manifolds with point singularities, Differ. Uravn., 48, 1612-1620, (2012) |

[2] | Savin, AYu; Sternin, BYu, Elliptic translators on manifolds with many-dimensional singularities, Differ. Uravn., 49, 513-527, (2013) |

[3] | Sternin, BYu, Elliptic and parabolic problems on manifolds with a boundary consisting of components of differential dimension, Tr. Mosk. Mat. Obs., 15, 346-382, (1966) · Zbl 0161.08504 |

[4] | Sternin, BYu, Relative elliptic theory and S. L. sobolev’s problem, Dokl. Akad. Nauk SSSR, 230, 287-290, (1976) |

[5] | Sternin, B.Yu., Topologicheskie aspekty problemy S.L. Soboleva (Topological Aspects of S.L. Sobolev Problem), Moscow, 1971. |

[6] | Sternin, BYu; Shatalov, VE, Relative elliptic theory and the Sobolev problem, Mat. Sb., 187, 115-144, (1996) |

[7] | Nazaikinskii, V; Sternin, B; Gil, J (ed.); Krainer, Th (ed.); Witt, I (ed.), Relative elliptic theory, 495-560, (2004) · Zbl 1090.58016 |

[8] | Nazaikinskii, V., Savin, A., Schulze, B.-W., and Sternin, B., Elliptic Theory on Singular Manifolds, Boca Raton, 2005. · Zbl 1138.58310 |

[9] | Sternin, BYu, Sobolev type elliptic problems for subdomains with point singularities, Dokl. Akad. Nauk SSSR, 184, 782-785, (1969) |

[10] | Sternin, BYu, S.L. Sobolev type problems in the case of submanifolds with multidimensional singularities, Dokl. Akad. Nauk SSSR, 189, 732-735, (1969) |

[11] | Sternin, BYu, Elliptic morphisms (riggings of elliptic operators) for submanifolds with singularities, Dokl. Akad. Nauk SSSR, 200, 45-48, (1971) |

[12] | Sternin, B.Yu., Ellipticheskaya teoriya na kompaktnykh mnogoobraziyakh s osobennostyami (Elliptic Theory on Compact Manifolds with Singularities), Moscow: Inst. Elektron. Mashinostroen., 1974. |

[13] | Zelikin, MI; Sternin, BYu, A system of integral equations that arises in the problem of S. L. Sobolev, Sibirsk. Mat. Zh., 18, 97-102, (1977) · Zbl 0358.35028 |

[14] | Savin, AYu; Sternin, BYu, On the index of elliptic translators, Dokl. Akad. Nauk, 436, 443-447, (2011) |

[15] | Gel’fand, I.M. and Shilov, G.E., Obobshchennye funktsii. Vyp. 2. Prostranstva osnovnykh i obobshchennykh funktsii (Spaces of Fundamental and Generalized Functions. Generalized Functions), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1958. · Zbl 0091.11103 |

[16] | Novikov, SP; Sternin, BYu, Traces of elliptic operators on submanifolds and \(K\)-theory, Dokl. Akad. Nauk SSSR, 170, 1265-1268, (1966) |

[17] | Atiyah, MF; Bott, R, The index problem for manifolds with boundary, 175-186, (1964) |

[18] | Nikol’skii, SM, Boundary properties of functions defined on a domain with corners. II. harmonic functions on rectangular domains, Mat. Sb., 43, 127-144, (1966) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.