Calculus of Variations and Geometric Measure Theory

G. Franzina

Non-local Torsion functions and Embeddings

created by franzina on 23 Jan 2018
modified on 07 May 2021


Published Paper

Inserted: 23 jan 2018
Last Updated: 7 may 2021

Journal: Applicable Analysis
Year: 2018


Given $s\in(0,1)$, we discuss the embedding of $\mathcal{D}_0^{s,p}(\Omega)$ in $L^q(\Omega)$. In particular, for $1\le q<p$ we deduce its compactness on all open sets $\Omega\subset \mathbb R^N$ 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 $\Omega$ in a suitable weak sense, for every open set $\Omega$. The proofs make use of a non-local Hardy-type inequality in $\mathcal{D}_0^{s,p}(\Omega)$, involving the fractional torsion function as a weight.