This is an announcement for the paper "Sequential lower semi-continuity of non-local functionals" by Peter Elbau.
Abstract: We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which depends on the function values at two different points. The characterisation we get is essentially that the density has to be convex in one variable if we integrate over the other one with an arbitrary test function in it. Moreover, we show that this condition is in the case of non-local functionals on real-valued Lebesgue spaces (up to some equivalence in the density) equivalent to the separate convexity of the density.
Archive classification: math.FA
Mathematics Subject Classification: 49J05, 49J45
Remarks: 23 pages
Submitted from: elbau@math.ethz.ch
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1104.2686
or