We characterize lower semicontinuity of integral functionals with respect to weak convergence in , 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].
DOI : 10.1051/cocv/2014036
Keywords: Lower semicontinuity, BV, quasiconvexity, free boundary
Benešová, Barbora 1 ; Krömer, Stefan 2 ; Kružík, Martin 3, 4
@article{COCV_2015__21_2_513_0,
author = {Bene\v{s}ov\'a, Barbora and Kr\"omer, Stefan and Kru\v{z}{\'\i}k, Martin},
title = {Boundary effects and weak$^{\star{}}$ lower semicontinuity for signed integral functionals on $BV$},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {513--534},
year = {2015},
publisher = {EDP Sciences},
volume = {21},
number = {2},
doi = {10.1051/cocv/2014036},
zbl = {1318.49022},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2014036/}
}
TY - JOUR
AU - Benešová, Barbora
AU - Krömer, Stefan
AU - Kružík, Martin
TI - Boundary effects and weak$^{\star{}}$ lower semicontinuity for signed integral functionals on $BV$
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2015
SP - 513
EP - 534
VL - 21
IS - 2
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/cocv/2014036/
DO - 10.1051/cocv/2014036
LA - en
ID - COCV_2015__21_2_513_0
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%A Krömer, Stefan
%A Kružík, Martin
%T Boundary effects and weak$^{\star{}}$ lower semicontinuity for signed integral functionals on $BV$
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 513-534
%V 21
%N 2
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2014036/
%R 10.1051/cocv/2014036
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%F COCV_2015__21_2_513_0
Benešová, Barbora; Krömer, Stefan; Kružík, Martin. Boundary effects and weak$^{\star{}}$ lower semicontinuity for signed integral functionals on $BV$. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 513-534. doi: 10.1051/cocv/2014036
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