On the total variations for the inverse of a \(BV\)-homeomorphism (Q2842998)

Let \({\Omega,\Omega'\subset\mathbb{R}^n}\) be domains and \({f=(u_1(x_1,\dots,x_n),\dots,u_n(x_1,\dots,x_n)):\Omega\to\Omega'}\) a homeomorphism whose coordinate functions \(u_i\), \(i=1,\dots,n\) are locally of bounded variations (i.e., \(f\in BV_{\mathrm{loc}}(\Omega)\)). By a result due to \textit{S. Hencl, P. Koskela} and \textit{J. Onninen} [Arch. Ration. Mech. Anal. 186, No. 3, 351--360 (2007; Zbl 1155.26007)] for the case \(n=2\) the coordinate functions \(x_1\), \(x_2\) of \(f^{-1}\) are also locally of bounded variation. In the article at hand, the authors present a different proof of this theorem with precise formulas for the total variations of the functions \(x_1\) and \(x_2\).NEWLINENEWLINENEWLINEIf the homeomorphism \(f\) is from the Sobolev class \(W_{\mathrm{loc}}^{1,n-1}(\Omega)\) then by a theorem due to \textit{M.~Csornyei, S.~Hencl} and \textit{J.~Maly} [J. Reine Angew. Math. 644, 221--235 (2010; Zbl 1210.46023)] its inverse is also in \({BV_{\mathrm{loc}}(\Omega')}\). Here the authors improve this theorem giving also precise formulas for the total variations of the functions \(x_i\), \(i=1,\dots,n\).NEWLINENEWLINENEWLINEOther results of the paper deal with properties of the weak limit of sequences of homeomorphisms from Sobolev classes and related questions.