Calculus of Variations and Geometric Measure Theory

E. Di Nezza - G. Palatucci - E. Valdinoci

Hitchhiker's guide to the fractional Sobolev spaces

created by palatucci on 21 Apr 2011
modified on 23 Jul 2012


Published Paper

Inserted: 21 apr 2011
Last Updated: 23 jul 2012

Journal: Bull. Sci. math.
Volume: 136
Number: 5
Pages: 521-573
Year: 2012


This paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.

Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces.

We also present some counterexamples in non-Lipschitz domains.

Keywords: nonlocal problems, fractional Laplacian, fractional Sobolev spaces, Gagliardo norms