Oscillations and concentrations in sequences of gradients

Kałamajska, Agnieszka; Kružík, Martin

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 $L^p(Ø;{ℝ}^{m\times n})$ 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.

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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

[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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

[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