Boundary behavior of Orlicz-Sobolev classes (Q2342007)

The main result of this paper concerns the extension of a homeomorphism defined on a domain to its closure. The necessary conditions split in conditions on the boundary of the domain and more importantly on the function. Besides being a homeomorphism, the conditions are concerned with being in the ``right'' Orlicz-Sobolev space and an integrability condition for the outer dilatation. On the way of proving the main theorem, the authors obtain results concerning differentiability almost everywhere, Luzin's condition~(N), estimates on the size of the image of sets, and a condition for being a \(Q\)-homeomorphism. Let me give some more references for the reader who is interested in results similar to some of the side results. Concerning the almost everywhere differentiability of continuous Sobolev mappings, there is also a result by \textit{E. M. Stein} [Ann. Math. (2) 113, 383--385 (1981; Zbl 0531.46021)] in the language of Lorentz spaces; see [\textit{J. Kauhanen} et al., Manuscr. Math. 100, No. 1, 87--101 (1999; Zbl 0976.26004)] (which gives also an alternative proof of Stein's result) and [\textit{J. Malý} et al., Stud. Math. 190, No. 1, 33--71 (2009; Zbl 1176.46033)] for the connection between Orlicz and Lorentz spaces. Even more in the spirit of the paper under review is \textit{V. Tengvall}'s paper [Calc.\ Var.\ Partial Differ.\ Equ. 51, No. 1--2, 381--399 (2014; Zbl 1305.26030)]. \textit{J. Onninen} studied the differentiability of monotone Sobolev mappings in [Real Anal. Exch. 26, No. 2, 761--772 (2001; Zbl 1055.46021)]. Luzin's condition~(N) is also studied in the paper by J.~Kauhanen et al. [loc. cit.]. A result about homeomorphic Sobolev mappings satisfying the \((n-1)\)-dimensional Luzin condition on hyperplanes can be found in [\textit{M. Csörnyei} et al., J. Reine Angew. Math. 644, 221--235 (2010; Zbl 1210.46023)].