Non-local torsion functions and embeddings

ABSTRACT

Given $s\in (0,1)$, we discuss the embedding of ${\mathcal{D}}_0^{s,p}(\mathrm{\Omega })$ in $L^q(\mathrm{\Omega })$. In particular, for $1\le q<p$ we deduce its compactness on all open sets $\mathrm{\Omega }\subset {\mathbb{R}}^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\mathrm{\Omega }$ in a suitable weak sense, for every open set $\mathrm{\Omega }$. The proofs make use of a non-local Hardy-type inequality in ${\mathcal{D}}_0^{s,p}(\mathrm{\Omega })$, involving the fractional torsion function as a weight.

KEYWORDS:

• Sobolev embedding
• torsional rigidity
• Hardy inequality
• non-local equations

AMS Subject Classifications:

• 35P15
• 46E35
• 34K37

Acknowledgements

The author wishes to thank Prof. Lorenzo Brasco for his useful comments on a preliminary version of the present manuscript, as well as for suggesting the problem, in Osaka in May 2017, during the Workshop ‘Geometric Properties for Parabolic and Elliptic PDEs’, the organisers of which are also gratefully acknowledged.

Notes

No potential conflict of interest was reported by the authors.

1 It suffices to apply inequality of [8, Proposition 3.5] with $\beta =1$, $\delta =0$, $F\equiv 1$, $L=1$, and ${\mathrm{\Omega }}^{\prime }=B_{2R}$.

2 In fact, that estimate implies (5.14(5.14) $$\begin{array}{c}\hfill {\Vert w\Vert }_{L^{\infty }(B_{R/4})}\le C_4(N,s,p)\left[{\left(-{\int }_{B_{R/2}}{w}^p\mathrm{d}x\right)}^{\frac{1}{p}}+(1+\delta \mathrm{Tail}(w,0,\frac{R}{4}))R^{\frac{\mathit{sp}}{p-1}}\right],\end{array}$$(5.14) ) with $\delta =1$, but a close inspection of its proof at scale 1 reveals that minor arrangements allow for the parameter $\delta$ to appear.

Additional information

Funding

This research is supported by the INdAM FOE 2014 [grant ‘SIES’].