Geodesic
Boundary effects and weak ☆ lower semicontinuity for signed integral functionals on BV
Benešová, Barbora1 ; Krömer, Stefan2 ; Kružík, Martin3, 4
1Department of Mathematics I, RWTH Aachen University, 52056 Aachen, Germany.
2Math. Inst., Universität zu Köln, 50923 Köln, Germany.
3Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic.
4Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16629 Praha 6, Czech Republic.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 513-534

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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].

Reçu le :
DOI : 10.1051/cocv/2014036
Classification : 49J45, 26B30, 52A99
Keywords: Lower semicontinuity, BV, quasiconvexity, free boundary

Benešová, Barbora  1   ; Krömer, Stefan  2   ; Kružík, Martin  3 , 4

1 Department of Mathematics I, RWTH Aachen University, 52056 Aachen, Germany.
2 Math. Inst., Universität zu Köln, 50923 Köln, Germany.
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. 
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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     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 = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014036/}
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