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A necessary condition for Sobolev extension domains in higher dimensions
- Publication Year :
- 2022
-
Abstract
- We give a necessary condition for a domain to have a bounded extension operator from $L^{1,p}(\Omega)$ to $L^{1,p}(\mathbb R^n)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of $\Omega$ integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for $W^{1,1}$-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain that is homeomorphic to a ball and has $n$-dimensional boundary.<br />Comment: 30 pages, 3 figures
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2207.00541
- Document Type :
- Working Paper