Calculus of Variations and Geometric Measure Theory

A. Tyulenev - S. Vodop'yanov

Sobolev $W^{1}_{p}$-spaces on $d$-thick closed subsets of $\mathbb{R}^{n}$

created by tyulenev on 30 Nov 2021


Published Paper

Inserted: 30 nov 2021
Last Updated: 30 nov 2021

Journal: Sbornik:Mathematics
Volume: 211
Number: 6
Pages: 786-837
Year: 2020


Let $S \subset \mathbb{R}^{n}$ be a nonempty closed set such that for some $d \in [0,n]$ and $\varepsilon > 0$ the d-Hausdorff content $\mathcal{H}^{d}_{\infty}(Q(x,r) \cap S) \geq \varepsilon r^{d}$ for all cubes $Q(x,r)$ with centre $x\in S$ and edge length $2r ∈ (0, 2]$. For each $p > \max{1, n − d}$ we give an intrinsic characterization of the trace space $W^{1}_{p}(\mathbb{R}^{n})
_{S}$ of the Sobolev space $W^{1}_{p}(\mathbb{R}^{n})$ to the set $S$. Furthermore, we prove the existence of a bounded linear operator $Ext:W^{1}_{p}(\mathbb{R}^{n})
_{S} \to W^{1}_{p}(\mathbb{R}^{n})$ such that $Ext$ is the right inverse to the standard trace operator. Our results extend those available in the case $p \in (1,n]$ for Ahlfors-regular sets $S$.

Keywords: Sobolev spaces, Whitney problem, traces, extension operators