Oscillations and concentrations in sequences of gradients
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 71-104.

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla {u}_{k}\right\}$, bounded in ${L}^{p}\left(Ø;{ℝ}^{m×n}\right)$ if $p>1$ and $\Omega \subset {ℝ}^{n}$ is a bounded domain with the extension property in ${W}^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega$ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

DOI : https://doi.org/10.1051/cocv:2007051
Classification : 49J45,  35B05
Mots clés : sequences of gradients, concentrations, oscillations, quasiconvexity
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Kružík, Martin; Kałamajska, Agnieszka. Oscillations and concentrations in sequences of gradients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 71-104. doi : 10.1051/cocv:2007051. http://www.numdam.org/articles/10.1051/cocv:2007051/

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | MR 751305 | Zbl 0565.49010

[2] J.J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997) 125-145. | EuDML 225088 | MR 1459885 | Zbl 0981.49012

[3] W. Allard, On the first variation of a varifold. Ann. Math. 95 (1972) 417-491. | MR 307015 | Zbl 0252.49028

[4] F.J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. 87 (1968) 321-391. | MR 225243 | Zbl 0162.24703

[5] 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. | MR 1036070 | Zbl 0991.49500

[6] J.M Ball and F. Murat, ${W}^{1,p}$-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR 759098 | Zbl 0549.46019

[7] P. Billingsley, Probability and Measure1995). | MR 1324786 | Zbl 0822.60002

[8] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer, Berlin (1989). | MR 990890 | Zbl 0703.49001

[9] 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. | MR 877643 | Zbl 0626.35059

[10] N. Dunford and J.T. Schwartz, Linear Operators, Part I. Interscience, New York (1967). | Zbl 0084.10402

[11] R. Engelking, General topology. 2nd edn., PWN, Warszawa (1985). | Zbl 0373.54002

[12] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992). | MR 1158660 | Zbl 0804.28001

[13] I. Fonseca, Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh 120A (1992) 95-115. | MR 1149987 | Zbl 0757.49013

[14] I. Fonseca and S. Müller, P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | MR 1617712 | Zbl 0920.49009

[15] L. Greco, T. Iwaniec and U. Subramanian, Another approach to biting convergence of Jacobians. Illin. Journ. Math. 47 (2003) 815-830. | MR 2007238 | Zbl 1060.46023

[16] M. De Guzmán, Differentiation of integrals in ${ℝ}^{n}$, Lecture Notes in Math. 481. Springer, Berlin (1975). | MR 457661 | Zbl 0327.26010

[17] O.M Hafsa, J.-P. Mandallena and G. Michaille, Homogenization of periodic nonconvex integral functionals in terms of Young measures. ESAIM: COCV 12 (2006) 35-51. | Numdam | MR 2192067 | Zbl 1107.49013

[18] A. Kałamajska, On lower semicontinuity of multiple integrals. Coll. Math. 74 (1997) 71-78. | MR 1455456 | Zbl 0893.49009

[19] A. Kałamajska, On Young measures controlling discontinuous functions. J. Conv. Anal. 13 (2006) 177-192. | MR 2211811 | Zbl 1109.49016

[20] A. Kałamajska and M. Kružík, On weak lower semicontinuity of integral functionals along concentrating sequences (in preparation).

[21] D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rat. Mech.Anal. 115 (1991) 329-365. | MR 1120852 | Zbl 0754.49020

[22] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | MR 1274138 | Zbl 0808.46046

[23] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mat-report 1994-34, Math. Institute, Technical University of Denmark (1994).

[24] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653-710. | MR 1686943 | Zbl 0924.49012

[25] M. Kružík and T. Roubíček, On the measures of DiPerna and Majda. Mathematica Bohemica 122 (1997) 383-399. | MR 1489400 | Zbl 0902.28009

[26] 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. | Zbl 0940.49002

[27] C. Licht, G. Michaille and S. Pagano, A model of elastic adhesive bonded joints through oscillation-concentration measures. Prépublication of the Institut de Mathématiques et de Modélisation de Montpellier, UMR-CNRS 5149. | Zbl 1118.49009

[28] P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1-28. | MR 788671 | Zbl 0573.49010

[29] V.G. Mazja, Sobolev Spaces. Springer, Berlin (1985). | MR 817985

[30] N.G. Meyers, Quasi-convexity and lower semicontinuity of multiple integrals of any order. Trans. Am. Math. Soc. 119 (1965) 125-149. | MR 188838 | Zbl 0166.38501

[31] C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966). | Zbl 0142.38701

[32] S. Müller, Variational models for microstructure and phase transisions. Lecture Notes in Mathematics 1713 (1999) 85-210. | MR 1731640 | Zbl 0968.74050

[33] P. Pedregal, Parametrized Measures and Variational Principles. Birkäuser, Basel (1997). | MR 1452107 | Zbl 0879.49017

[34] Yu.G. Reshetnyak, The generalized derivatives and the a.e. differentiability. Mat. Sb. 75 (1968) 323-334 (in Russian). | MR 225159 | Zbl 0165.47202

[35] Yu.G. Reshetnyak, Weak convergence and completely additive vector functions on a set. Sibirsk. Mat. Zh. 9 (1968) 1039-1045. | Zbl 0176.44402

[36] T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997). | MR 1458067 | Zbl 0880.49002

[37] M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations. Comm. Part. Diff. Equa. 7 (1982) 959-1000. | MR 668586 | Zbl 0496.35058

[38] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton university Press, Princeton (1970). | MR 290095 | Zbl 0207.13501

[39] 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 Math. 39, San Francisco (1979). | MR 584398 | Zbl 0437.35004

[40] 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 et al. Eds., Proceedings of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, September 3-9, 2000, Springer, Berlin (2002). | MR 1998790 | Zbl 1015.35001

[41] M. Valadier, Young measures, in Methods of Nonconvex Analysis, A. Cellina Ed., Lect. Notes Math. 1446, Springer, Berlin (1990) 152-188. | MR 1079763 | Zbl 0738.28004

[42] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | MR 372708 | Zbl 0253.49001

[43] 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 III 30 (1937) 212-234. | JFM 63.1064.01

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