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 \(n\).


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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