DiPerna’s and Majda’s generalization of Young measures is used to describe oscillations and concentrations in sequences of maps satisfying a linear differential constraint . Applications to sequential weak lower semicontinuity of integral functionals on -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det in measures on the closure of if in . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets must be removed to obtain weak lower semicontinuity of along . Specifically, are arbitrarily thin “boundary layers”.
Mots clés : concentrations, oscillations, Young measures
@article{COCV_2010__16_2_472_0, author = {Fonseca, Irene and Kru\v{z}{\'\i}k, Martin}, title = {Oscillations and concentrations generated by ${\mathcal {A}}$-free mappings and weak lower semicontinuity of integral functionals}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {472--502}, publisher = {EDP-Sciences}, volume = {16}, number = {2}, year = {2010}, doi = {10.1051/cocv/2009006}, mrnumber = {2654203}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2009006/} }
TY - JOUR AU - Fonseca, Irene AU - Kružík, Martin TI - Oscillations and concentrations generated by ${\mathcal {A}}$-free mappings and weak lower semicontinuity of integral functionals JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 472 EP - 502 VL - 16 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009006/ DO - 10.1051/cocv/2009006 LA - en ID - COCV_2010__16_2_472_0 ER -
%0 Journal Article %A Fonseca, Irene %A Kružík, Martin %T Oscillations and concentrations generated by ${\mathcal {A}}$-free mappings and weak lower semicontinuity of integral functionals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 472-502 %V 16 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009006/ %R 10.1051/cocv/2009006 %G en %F COCV_2010__16_2_472_0
Fonseca, Irene; Kružík, Martin. Oscillations and concentrations generated by ${\mathcal {A}}$-free mappings and weak lower semicontinuity of integral functionals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 472-502. doi : 10.1051/cocv/2009006. http://www.numdam.org/articles/10.1051/cocv/2009006/
[1] Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997) 125-145. | Zbl
and ,[2] A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transition, M. Rascle, D. Serre and M. Slemrod Eds., Lect. Notes Phys. 344, Springer, Berlin (1989) 207-215. | Zbl
,[3] W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | Zbl
and ,[4] Lower semicontinuity of multiple integrals and the biting lemma. Proc. Roy. Soc. Edinb. A 114 (1990) 367-379. | Zbl
and ,[5] A-quasiconvexity: relaxation and homogenization. ESAIM: COCV 5 (2000) 539-577. | Numdam | Zbl
, and ,[6] Continuity and compactness in measure. Adv. Math. 37 (1980) 16-26. | Zbl
and ,[7] Direct Methods in the Calculus of Variations. Springer, Berlin (1989). | Zbl
,[8] Energy minimizers for large ferromagnetic bodies. Arch. Rat. Mech. Anal. 125 (1993) 99-143. | Zbl
,[9] Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108 (1987) 667-689. | Zbl
and ,[10] Linear Operators, Part I. Interscience, New York (1967). | Zbl
and ,[11] General topology. Second Edition, PWN, Warszawa (1985). | Zbl
,[12] Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992). | Zbl
and ,[13] Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinb. A 120 (1992) 95-115. | Zbl
,[14] Modern Methods in the Calculus of Variations: Lp Spaces. Springer (2007). | Zbl
and ,[15] -quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl
and ,[16] Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | Zbl
, and ,[17] Global higher integrability of Jacobians on bounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 193-217. | Numdam | Zbl
, , and ,[18] Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71-104. | Numdam | Zbl
and ,[19] Characterization of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329-365. | Zbl
and ,[20] Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23 (1992) 1-19. | Zbl
and ,[21] Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | Zbl
and ,[22] Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653-710. | Zbl
,[23] Explicit characterization of Lp-Young measures. J. Math. Anal. Appl. 198 (1996) 830-843. | Zbl
and ,[24] On the measures of DiPerna and Majda. Mathematica Bohemica 122 (1997) 383-399. | Zbl
and ,[25] Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim. 20 (1999) 511-530. | Zbl
and ,[26] A model of elastic adhesive bonded joints through oscillation-concentration measures. J. Math. Pures Appl. 87 (2007) 343-365. | Zbl
, and ,[27] Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1-28. | Zbl
,[28] Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
,[29] Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math. 412 (1990) 20-34. | Zbl
,[30] Variational models for microstructure and phase transisions. Lect. Notes Math. 1713 (1999) 85-210. | Zbl
,[31] Relaxation in ferromagnetism: the rigid case, J. Nonlinear Sci. 4 (1994) 105-125. | Zbl
,[32] Parametrized Measures and Variational Principles. Birkäuser, Basel (1997). | Zbl
,[33] Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997). | Zbl
,[34] Convergence of solutions to nonlinear dispersive equations. Comm. Partial Diff. Eq. 7 (1982) 959-1000. | Zbl
,[35] Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics, R.J. Knops Ed., Heriott-Watt Symposium IV, Pitman Res. Notes in Math. 39, San Francisco (1979). | Zbl
,[36] Mathematical tools for studying oscillations and concentrations: From Young measures to H-measures and their variants, in Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives, N. Antonič, C.J. Van Duijin and W. Jager Eds., Proceedings of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, September 3-9, 2000, Springer, Berlin (2002). | Zbl
,[37] Young measures, in Methods of Nonconvex Analysis, A. Cellina Ed., Lect. Notes Math. 1446, Springer, Berlin (1990) 152-188. | Zbl
,[38] Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | Zbl
,[39] Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe III 30 (1937) 212-234. | JFM | Zbl
,Cité par Sources :