We characterize lower semicontinuity of integral functionals with respect to weak${}^{\u2606}$ 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].

DOI: 10.1051/cocv/2014036

Keywords: Lower semicontinuity, BV, quasiconvexity, free boundary

^{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}, publisher = {EDP-Sciences}, volume = {21}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014036}, zbl = {1318.49022}, language = {en}, url = {http://archive.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 - http://archive.numdam.org/articles/10.1051/cocv/2014036/ DO - 10.1051/cocv/2014036 LA - en ID - COCV_2015__21_2_513_0 ER -

%0 Journal Article %A Benešová, Barbora %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 http://archive.numdam.org/articles/10.1051/cocv/2014036/ %R 10.1051/cocv/2014036 %G en %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, Volume 21 (2015) no. 2, pp. 513-534. doi : 10.1051/cocv/2014036. http://archive.numdam.org/articles/10.1051/cocv/2014036/

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