On equicontinuity of a family of space mappings in the closure of a domain
The present article is devoted to the study of mappings with finite distortion in \({\mathbb R}^n,\) \(n\geq 2\). It is shown that every family of \(Q\)-homeomorphisms defined in a domain \(D\) and satisfying some normalization conditions is equicontinuous in the closure of \(D\) provided that a majorant \(Q(x)\) has finite mean oscillation at every point of \(\overline{D},\) where the domain \(D\) is locally connected on the boundary and the boundary of \(D^{\,\prime}=f(D)\) is strongly accessible. Another sufficient condition providing the equicontinuity of the above family in \(\overline{D}\) requires that a majorant \(Q(x)\) has only logarithmic singularities of order at most \(n-1\) at every point of \(\overline{D}\) with the same conditions on \(D\) and \(D^{\,\prime}\). It is also shown that for every family of \(Q\)-homeomorphisms in a domain \(D\) satisfying the normalization conditions and such that \(Q(x)\in L_{\mathrm {loc}}^1,\) a family of inverse homeomorphisms is normal in \(D^{\,\prime}=f(D)\) provided that \(D\) is locally connected on the boundary and \(D^{\,\prime}\) is a so-called QED-domain.
scientific article; zbMATH DE number 6062230