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arXiv · 2207.00541

A necessary condition for Sobolev extension domains in higher dimensions

Miguel García-Bravo·Tapio Rajala·Jyrki Takanen
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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.

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Miguel García-Bravo, Tapio Rajala, Jyrki Takanen. 2022-07-01. A necessary condition for Sobolev extension domains in higher dimensions. https://arxiv.org/abs/2207.00541

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