Boundary effects and weak^⋆ lower semicontinuity for signed integral functionals on BV^∗

¹ Department of Mathematics I, RWTH Aachen University, 52056 Aachen, Germany.
² Math. Inst., Universität zu Köln, 50923 Köln, Germany.
skroemer@math.uni-koeln.de
³ Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic.
⁴ Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629 Praha 6, Czech Republic.

Received: 21 May 2014

Abstract

We characterize lower semicontinuity of integral functionals with respect to weak^⋆ convergence in BV, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition – quasi-sublinear growth from below at points of the boundary – which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by Kristensen and Rindler [J. Kristensen and F. Rindler, Arch. Rat. Mech. Anal. 197 (2010) 539–598; J. Kristensen and F. Rindler, Calc. Var. 37 (2010) 29–62].