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Failure of the condition N below \(W^{1,n}\). (English) Zbl 1027.26015
The Lusin condition N says that given a continuous mapping \(f\) from a bounded subset \(\Omega\) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), \(n\geq 2\), then \(f\) maps every subset of \(\Omega\) of zero measure to a set of zero measure. The well-known result by M. Marcus and V. Mizel [Bull. Am. Math. Soc. 79, 790-795 (1973; Zbl 0275.49041)] states that \(f\in W^{1,1}(\Omega,\mathbb{R}^n)\) satisfies the condition N under the assumption \(|Df|\in L^p(\Omega)\) for some \(p>n\). The condition N may fail if this assumption holds only for \(p\leq n\) and then there is a question about the size of an exceptional set, i.e., the set outside of which the condition holds. J. Malý and O. Martio [J. Reine Angew. Math. 458, 19-36 (1995; Zbl 0812.30007)] proved that for \(f\in W^{1,n}(\Omega,\mathbb{R}^n)\) exceptional sets have the Hausdorff dimension zero.
The author deals with the case below \(W^{1,n}(\Omega,\mathbb{R}^n)\). He proves that for \(Q_0=\left[0,1\right]^n\) and \(f\in W^{1,1}(Q_0,\mathbb{R}^n)\) such that \(\underset{0<\varepsilon\leq n-1} \sup \varepsilon\int_{Q_0}|Df(x)|^{n-\varepsilon}dx<\infty\) one cannot, in general, find an exceptional set of Hausdorff dimension smaller than \(n\).

MSC:
26B35 Special properties of functions of several variables, Hölder conditions, etc.
74B20 Nonlinear elasticity
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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