Calculus of Variations and Geometric Measure Theory
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P. Gladbach - H. Olbermann

Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

created by olbermann on 23 Jul 2019

[BibTeX]

preprint

Inserted: 23 jul 2019

Year: 2019

ArXiv: 1903.07420 PDF

Abstract:

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants $Ju$ for functions $u$ in fractional Sobolev spaces $W^{s,p}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. The weak forms of the formulae are proved for the range $sp>n-1$, $s\geq \frac{n-1}{n}$, while the strong versions are proved for the range $sp> n$, $s\geq \frac{n}{n+1}$. We also provide a chain rule for distributional Jacobian determinants of H\"older functions and point out its relation to two open problems in geometric analysis.

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