On stability of classes of Lipschitz mappings generated by compact sets of linear mappings.

*(English. Russian original)*Zbl 0959.47008
Sib. Math. J. 41, No. 4, 656-670 (2000); translation from Sib. Mat. Zh. 41, No. 4, 792-810 (2000).

The article is devoted to a further development of the stability theory of classes of Lipschitz mappings in the framework of the concept of \(\omega\)-stability. The main object under study is the class of mappings described below. Suppose that \(G\) is a nonempty compact subset of the space L\((\mathbb R^n,\mathbb R^m)\) of linear mappings from \(\mathbb R^n\) into \(\mathbb R^m\). The class of all locally Lipschitz mappings \(g\:\Delta\to\mathbb R^m\) of domains \(\Delta\subset\mathbb R^n\), for each of which there is a connected component \(K\) of \(G\) such that the differentials \(g'(x)\) at almost all points \(x\in\text{dom }g\) belong to \(K\), is said to be \((*)\)-generated by \(G\) and denoted by \(\mathfrak Z(G)\).

The author starts with proving the theorem claiming that if \(G\) is a compact convex set then the class \(\mathfrak Z(G)\) is \(\omega\)-stable. One of the main tools is the theorem about preservation of stability under set-theoretic operations on \(\omega\)-stable classes. As a result, the author completely solves the stability problem for the classes \(\mathfrak Z(G)\) when either of the dimensions \(n\) and \(m\) equals 1. Using these results, the author obtains theorems on stability of classes of Lipschitz solutions to systems of linear partial differential equations and a theorem on \(\omega\)-stability of classes of conformal mappings which may simultaneously contain sense-preserving and sense-reversing mappings.

The author finds a fruitful application of the notion of weak connectedness of sets in vector spaces which was introduced into consideration for studying the structure of the image of the derivative of a differentiable vector-valued mapping.

The author starts with proving the theorem claiming that if \(G\) is a compact convex set then the class \(\mathfrak Z(G)\) is \(\omega\)-stable. One of the main tools is the theorem about preservation of stability under set-theoretic operations on \(\omega\)-stable classes. As a result, the author completely solves the stability problem for the classes \(\mathfrak Z(G)\) when either of the dimensions \(n\) and \(m\) equals 1. Using these results, the author obtains theorems on stability of classes of Lipschitz solutions to systems of linear partial differential equations and a theorem on \(\omega\)-stability of classes of conformal mappings which may simultaneously contain sense-preserving and sense-reversing mappings.

The author finds a fruitful application of the notion of weak connectedness of sets in vector spaces which was introduced into consideration for studying the structure of the image of the derivative of a differentiable vector-valued mapping.

Reviewer: V.Grebenev (Novosibirsk)

##### MSC:

47A55 | Perturbation theory of linear operators |

47B07 | Linear operators defined by compactness properties |

47B65 | Positive linear operators and order-bounded operators |

47F05 | General theory of partial differential operators |

46G05 | Derivatives of functions in infinite-dimensional spaces |

46T20 | Continuous and differentiable maps in nonlinear functional analysis |

##### Keywords:

compact sets of linear mappings; Lipschitz mapping; preservation of stability under set-theoretic operations on \(\omega\)-stable classes; \(\omega \)-stability of classes of conformal mappings; linear partial differential equation
PDF
BibTeX
XML
Cite

\textit{M. V. Korobkov}, Sib. Math. J. 41, No. 4, 792--810 (2000; Zbl 0959.47008); translation from Sib. Mat. Zh. 41, No. 4, 792--810 (2000)

**OpenURL**

##### References:

[1] | Reshetnyak Yu. G., Stability Theorems in Geometry and Analysis [in Russian], Nauka, Novosibirsk (1982). · Zbl 0523.53025 |

[2] | John F., ”Rotation and strain,” Comm. Pure Appl. Math.,14, No. 3, 391–413 (1961). · Zbl 0102.17404 |

[3] | Kopylov A. P., ”On stability of isometric mappings,” Sibirsk. Mat. Zh.,25, No. 2, 132–144 (1984). · Zbl 0546.30019 |

[4] | Kopylov A. P., Stability in theC-Norm of Classes of Mappings [in Russian], Nauka, Novosibirsk, (1990). · Zbl 0772.30023 |

[5] | Egorov A. A., ”On stability of classes of affine mappings,” Sibirsk. Mat. Zh.,36, No. 5, 1080–1095 (1995). · Zbl 0855.53016 |

[6] | Egorov A. A., ”Stability of classes of Lipschitz continuous solutions to systems of first order linear differential equations,” Dokl. Akad. Nauk,356, No. 5, 583–587 (1997). |

[7] | Egorov A. A., ”On stability of classes of Lipschitz continuous solutions to systems of first order linear differential equations,” Sibirsk. Mat. Zh.,40, No. 3, 548–553 (1999). · Zbl 0928.35042 |

[8] | Korobkov M. V., ”On one generalization of connectedness and its application to differential calculus and stability theory,” Dokl. Akad. Nauk,363, No. 5, 590–593 (1998). · Zbl 0959.58006 |

[9] | Korobkov M. V., ”On one generalization of the Darboux theorem to the multidimensional case,” Sibirsk. Mat. Zh.,41, No. 1, 118–133 (2000). · Zbl 0963.26006 |

[10] | Bourbaki N., Varietétés Différentielles et Analytiques. Fascicule de Résultats [Russian translation], Mir, Moscow (1975). |

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.