Geodesic
Boundary effects and weak ☆ lower semicontinuity for signed integral functionals on BV
Benešová, Barbora1 ; Krömer, Stefan2 ; Kružík, Martin3, 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.
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. 
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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},
     publisher = {EDP-Sciences},
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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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