Technion - Israel Institute of Technology Department of Mathematics ===================================== PDE AND APPLIED MATHEMATICS SEMINAR ===================================== 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 ABSTRACT: 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 necessary. For further info: Itai Shafrir <shafrir@techunix.technion.ac.il> For past and future Applied Math/PDE seminars see: <http://www.math.technion.ac.il/pde/seminar.html> --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Itai Shafrir <shafrir@math.technion.ac.il>