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The co-area formula for Sobolev mappings. (English) Zbl 1034.46032

The co-area formula \[ \int\limits_\Omega g(x)| J_mf(x)| \, dx = \int_{\mathbb{R}^m}\int_{f^{-1}(y)}g(x) \mathcal{H}^{n-m}(x)\,dy \] where \(\Omega \subset \mathbb{R}^n\) is an open set, \(f:\Omega \rightarrow \mathbb{R}^m\), \(J_mf(x)\) is the \(m\)-dimensional Jacobian of \(f\) and \(g: \Omega \rightarrow \mathbb{R}\) is integrable and \(1\leq m<n\), which is due to H. Federer in the case of Lipschitzian \(f\), is extended to the case of Sobolev mappings \(f\in W_{\text{loc}}^{1,p}(\Omega; \mathbb{R}^m)\), where \(p>m\) or \(p\geq m=1\). The limiting case \(p=m\) for Hölder continuous mappings is also treated.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26B10 Implicit function theorems, Jacobians, transformations with several variables
26B35 Special properties of functions of several variables, Hölder conditions, etc.
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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