Equivalent Characterizations and Applications of Fractional Sobolev Spaces with Partially Vanishing Traces on $(\epsilon ,\delta ,D)$-Domains Supporting $D$-Adapted Fractional Hardy Inequalities

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be an $(\epsilon ,\delta ,D)$-domain, with $\epsilon \in (0,1]$, $\delta \in (0,\infty ]$, and $D\subset \partial \Omega$ being a closed part of $\partial \Omega$, which is a general open connected set when $D=\partial \Omega$ and an $(\epsilon ,\delta )$-domain when $D=\varnothing$. Let $s\in (0,1)$ and $p\in [1,\infty )$. If $W^{s,p}(\Omega )$, ${\mathcal{W}}^{s,p}(\Omega )$, and ${\mathring{W}}_D^{s,p}(\Omega )$ are the fractional Sobolev spaces on $\Omega$ that are defined respectively via the restriction of $W^{s,p}({\mathbb{R}}^n)$ to $\Omega$, the intrinsic Gagliardo norm, and the completion of all $C^{\infty }(\Omega )$ functions with compact support away from $D$, in this article we prove their equivalences [that is, $W^{s,p}(\Omega )={\mathcal{W}}^{s,p}(\Omega )={\mathring{W}}_D^{s,p}(\Omega )$] if $\Omega$ supports a $D$-adapted fractional Hardy inequality and, moreover, when $sp\ne 1$ such a fractional Hardy inequality is shown to be necessary to guarantee these equivalences under some mild geometric conditions on $\Omega$. Using the aforementioned equivalences, we show that the real interpolation space $(L^p(\Omega ),{\mathring{W}}_D^{1,p}(\Omega ){)}_{s,p}$ equals to some weighted fractional order Sobolev space ${\mathcal{W}}_{d_D^s}^{s,p}(\Omega )$ when $p\in (1,\infty )$. Applying this to the elliptic operator ${\mathcal{L}}_D$ in $\Omega$ with mixed boundary condition, we characterize both the domain of its fractional power and the parabolic maximal regularity of its Cauchy initial problem by means of ${\mathcal{W}}_{d_D^s}^{s,p}(\Omega )$.

42 pages; Submitted