# Oscillations and concentrations generated by $\mathcal{A}$-free mappings and weak lower semicontinuity of integral functionals

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 2, page 472-502
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topFonseca, Irene, and 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 16.2 (2010): 472-502. <http://eudml.org/doc/250738>.

@article{Fonseca2010,

abstract = {
DiPerna's and Majda's generalization of Young measures
is used to describe oscillations and concentrations in sequences of maps $\\{u_k\\}_\{k\in\{\mathbb N\}\} \subset L^p(\Omega;\{\mathbb R\}^m)$ satisfying a linear differential constraint $\{\mathcal A\}u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $\{\mathcal A\}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla\varphi_k\stackrel\{*\}\{\rightharpoonup\}\{\rm det\}\nabla\varphi$ in measures on the closure of $\Omega\subset\{\mathbb R\}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^\{1,n\}(\Omega;\{\mathbb R\}^n)$. This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma
precisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_\{\Omega\setminus\Omega_j\} v(u(x))\,\{\rm d\}x$ along $\\{u_k\\}\subset L^p(\Omega;\{\mathbb R\}^m)\cap\{\rm ker\}\ \{\mathcal A\}$. Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.
},

author = {Fonseca, Irene, Kružík, Martin},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Concentrations; oscillations; Young measures; concentrations},

language = {eng},

month = {4},

number = {2},

pages = {472-502},

publisher = {EDP Sciences},

title = {Oscillations and concentrations generated by $\{\mathcal A\}$-free mappings and weak lower semicontinuity of integral functionals},

url = {http://eudml.org/doc/250738},

volume = {16},

year = {2010},

}

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

DA - 2010/4//

PB - EDP Sciences

VL - 16

IS - 2

SP - 472

EP - 502

AB -
DiPerna's and Majda's generalization of Young measures
is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$. This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma
precisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$. Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.

LA - eng

KW - Concentrations; oscillations; Young measures; concentrations

UR - http://eudml.org/doc/250738

ER -

## References

top- J.J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures. J. Convex Anal.4 (1997) 125–145.
- J.M. Ball, 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.
- J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.
- J.M. Ball and K.-W. Zhang, Lower semicontinuity of multiple integrals and the biting lemma. Proc. Roy. Soc. Edinb. A114 (1990) 367–379.
- A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity: relaxation and homogenization. ESAIM: COCV5 (2000) 539–577.
- J.K. Brooks and R.V. Chacon, Continuity and compactness in measure. Adv. Math.37 (1980) 16–26.
- B. Dacorogna, Direct Methods in the Calculus of Variations. Springer, Berlin (1989).
- A. DeSimone, Energy minimizers for large ferromagnetic bodies. Arch. Rat. Mech. Anal.125 (1993) 99–143.
- R.J. DiPerna and A.J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys.108 (1987) 667–689.
- N. Dunford and J.T. Schwartz, Linear Operators, Part I. Interscience, New York (1967).
- R. Engelking, General topology. Second Edition, PWN, Warszawa (1985).
- L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992).
- I. Fonseca, Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinb. A120 (1992) 95–115.
- I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces. Springer (2007).
- I. Fonseca and S. Müller, $\mathcal{A}$-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal.30 (1999) 1355–1390.
- I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal.29 (1998) 736–756.
- J. Hogan, C. Li, A. McIntosh and K. Zhang, Global higher integrability of Jacobians on bounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire17 (2000) 193–217.
- A. Kałamajska and M. Kružík, Oscillations and concentrations in sequences of gradients. ESAIM: COCV14 (2008) 71–104.
- D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365.
- D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal.23 (1992) 1–19.
- D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal.4 (1994) 59–90.
- J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann.313 (1999) 653–710.
- M. Kružík and T. Roubíček, Explicit characterization of Lp-Young measures. J. Math. Anal. Appl.198 (1996) 830–843.
- M. Kružík and T. Roubíček, On the measures of DiPerna and Majda. Mathematica Bohemica122 (1997) 383–399.
- M. Kružík and T. Roubíček, Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim.20 (1999) 511–530.
- C. Licht, G. Michaille and S. Pagano, A model of elastic adhesive bonded joints through oscillation-concentration measures. J. Math. Pures Appl.87 (2007) 343–365.
- P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math.51 (1985) 1–28.
- C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
- S. Müller, Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math.412 (1990) 20–34.
- S. Müller, Variational models for microstructure and phase transisions. Lect. Notes Math.1713 (1999) 85–210.
- P. Pedregal, Relaxation in ferromagnetism: the rigid case, J. Nonlinear Sci.4 (1994) 105–125.
- P. Pedregal, Parametrized Measures and Variational Principles. Birkäuser, Basel (1997).
- T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997).
- M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations. Comm. Partial Diff. Eq.7 (1982) 959–1000.
- L. Tartar, 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).
- L. Tartar, 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).
- M. Valadier, Young measures, in Methods of Nonconvex Analysis, A. Cellina Ed., Lect. Notes Math.1446, Springer, Berlin (1990) 152–188.
- J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972).
- L.C. Young, 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 III30 (1937) 212–234.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.