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
Mots-clés : Lower semicontinuity, BV, quasiconvexity, free boundary
@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, Tome 21 (2015) no. 2, pp. 513-534. doi : 10.1051/cocv/2014036. http://archive.numdam.org/articles/10.1051/cocv/2014036/
Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125–145. | DOI | Zbl
and ,L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Clarendon Press, Oxford, 2000. | Zbl
Lower semicontinuity and relaxation of signed functionals with linear growth in the context of -quasiconvexity. Calc. Var. Partial Differ. Equ. 47 (2013) 465–498. | DOI | Zbl
, , and ,Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86 (1984) 251–277. | DOI | Zbl
and ,On the Dirichlet problem for variational integrals in . J. Reine Angew. Math. 674 (2013) 113–194. | Zbl
and ,Quasi-convex integrands and lower semicontinuity in . SIAM J. Math. Anal. 23 (1992) 1081–1098. | DOI | Zbl
and ,Relaxation of quasiconvex functionals in for integrands . Arch. Ration. Mech. Anal. 123 (1993) 1–49. | DOI | Zbl
and ,Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. | DOI | Zbl
, and ,Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71–104. | Numdam | Zbl
and ,Sequential weak continuity of null lagrangians at the boundary. Calc. Var. Partial Differ. Equ. 49 (2014) 1263–1278. | DOI | Zbl
, and ,Characterization of generalized gradient Young measures generated by sequences in and BV. Arch. Ration. Mech. Anal. 197 (2010) 539–598. | DOI | Zbl
and ,J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mat-report 1994-34, Math. Institute, Technical University of Denmark, 1994.
Relaxation of signed integral functionals in BV. Calc. Var. Partial Differ. Equ. 37 (2010) 29–62. | DOI | Zbl
and ,Oscillations and concentrations in sequences of gradients up to the boundary. J. Convex Anal. 20 (2013) 723–752. | Zbl
and ,On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals. Adv. Calc. Var. 3 (2010) 387–408. | Zbl
,Quasiconvexity at the boundary and concentration effects generated by gradients. ESAIM: COCV 19 (2013) 679–700. | Numdam | Zbl
,Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition. J. Elasticity 51 (1998) 23–41. | DOI | Zbl
and ,Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25–53. | DOI | Zbl
,F. Rindler and G. Shaw, Strictly continuous extensions and convex lower semicontinuity of functionals with linear growth. Preprint arXiv:1312.4554v2 [math.AP] (2013).
P. Sprenger, Quasikonvexität am Rande und Null-Lagrange-Funktionen in der nichtkonvexen Variationsrechnung. Ph.D. thesis, Universität Hannover (1996). | Zbl
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