Technion - Israel Institute of Technology
          Department of Mathematics
DATE: Tuesday, October 30, 2012
SPEAKER: Daniel Spector, Technion
TITLE: Asymptotic Characterization of Sobolev spaces for Arbitrary Open Sets
PLACE: Room 814, Amado Mathematics Building, Technion
TIME: 14:30
The 2001 paper of Bourgain, Brezis, and Mironescu "Another Look
at Sobolev Spaces" established a characterization of Sobolev spaces for
smooth, bounded domains via the asymptotics of non-local functionals.  The
ideas introduced in this paper inaugurated the study of these "non-local"
quantities and raised a number of questions to be answered (in fact, since
these quantities are defined by the asymptotics of non-local functionals,
they are, as we will see, in some sense local).  The techniques introduced
relied on extending Sobolev functions, which depends on assuming
sufficient boundary regularity, and so among these questions is the
problem of finding a similar non-local quantity that characterizes the
Sobolev spaces for an arbitrary open set.  This is mentioned in the
literature in two conjectures that I am aware of, each giving a possible
condition.  In this talk I will discuss the original motivation for the
study of these non-local quantities, recent work done in collaboration
with Giovanni Leoni giving a necessary and sufficient condition
characterizing Sobolev spaces on an arbitrary open set with a non-local
quantity, and some counterexamples that show though the assumptions of
conjectures are sufficient (something I will establish), they are not
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