The area formula for \(W^{1,n}\)-mappings. (English) Zbl 0812.30006

Malý, Jan

Commentat. Math. Univ. Carol. 35, No. 2, 291-298 (1994).

The area formula (equivalent to the change of variables formula) \[ \int_ S | J(x, f)| dx= \int_{\mathbb{R}^ n} N(y, f, S) dy \] does not hold for continuous \(W^{1,p}\)-mappings \(f: \Omega\to \mathbb{R}^ n\), \(p\leq n\). Here \(\Omega\) is an open set in \(\mathbb{R}^ n\), \(S\subset \Omega\) is measurable and \(N(y, f, S)= \#\{x\in S: f(x)= y\}\) is the Banach indicatrix. This is due to the failure of the Lusin property (N) for these mappings. The author looks for special sets \(S\) for which the area formula holds and proves that if \(f\) is a quasi-continuous representative of a \(W^{1,n}\)-mapping and if \(S\) is the set of all points at which \(f\) is approximately Hölder continuous, then the area formula holds in \(S\). He also shows that \(\Omega\backslash S\) has Hausdorff dimension zero. The proof makes use of the Gehring oscillation lemma for \(W^{1,n}\)-mappings; this is employed to prove an oscillation result which shows that in each ball the oscillation of a \(W^{1,n}\)- mapping can be controlled by its \(n\)-Dirichlet integral except for a set of small measure.

Reviewer: O.Martio (Helsinki)

MSC:

30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26B15	Integration of real functions of several variables: length, area, volume
26B20	Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
28A78	Hausdorff and packing measures

Keywords:

area formula; change of variables formula; Lusin property (N); \(W^{1,n}\)-mapping; Gehring oscillation lemma; \(n\)-Dirichlet integral