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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.
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
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