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

TL;DR

This paper establishes the equivalence of different fractional Sobolev spaces on special domains supporting Hardy inequalities, and applies these results to characterize fractional powers of elliptic operators with mixed boundary conditions.

Contribution

It proves the equivalence of various fractional Sobolev spaces on $( ext{ extepsilon}, ext{ extdelta},D)$-domains under Hardy inequality support, and applies this to operator theory and boundary value problems.

Findings

Proves equivalence of fractional Sobolev spaces under Hardy inequality conditions.

Characterizes domain of fractional elliptic operators with mixed boundary conditions.

Links interpolation spaces to weighted fractional Sobolev spaces.

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 $D=\emptyset$. 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 $\mathcal{W}^{s,p}_{d_D^s}(\Omega)$ 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 $\mathcal{W}^{s,p}_{d_D^s}(\Omega)$.