Boundary effects and weak$^{\star{}}$ lower semicontinuity for signed integral functionals on $BV$

Benešová, Barbora; Krömer, Stefan; Kružík, Martin

doi:10.1051/cocv/2014036

Boundary effects and weak

^{☆}

lower semicontinuity for signed integral functionals on

B V

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.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 513-534.

Résumé

We characterize lower semicontinuity of integral functionals with respect to weak $^{☆}$ convergence in $B V$ , 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 : 2014-05-21

Zbl

DOI : 10.1051/cocv/2014036

Classification : 49J45, 26B30, 52A99
Mots clés : Lower semicontinuity, BV, quasiconvexity, free boundary

Affiliations des auteurs :

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

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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},
     volume = {21},
     number = {2},
     year = {2015},
     doi = {10.1051/cocv/2014036},
     zbl = {1318.49022},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2014036/}
}

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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. http://www.numdam.org/articles/10.1051/cocv/2014036/

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