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 ${L}^{p}(\xd8;{\mathbb{R}}^{m\times n})$ if $p>1$ and $\Omega \subset {\mathbb{R}}^{n}$ 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.

Keywords: sequences of gradients, concentrations, oscillations, quasiconvexity

@article{COCV_2008__14_1_71_0, author = {Ka{\l}amajska, Agnieszka and Kru\v{z}{\'\i}k, Martin}, title = {Oscillations and concentrations in sequences of gradients}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {71--104}, publisher = {EDP-Sciences}, volume = {14}, number = {1}, year = {2008}, doi = {10.1051/cocv:2007051}, mrnumber = {2375752}, zbl = {1140.49009}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007051/} }

TY - JOUR AU - Kałamajska, Agnieszka AU - Kružík, Martin TI - Oscillations and concentrations in sequences of gradients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 71 EP - 104 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007051/ DO - 10.1051/cocv:2007051 LA - en ID - COCV_2008__14_1_71_0 ER -

%0 Journal Article %A Kałamajska, Agnieszka %A Kružík, Martin %T Oscillations and concentrations in sequences of gradients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 71-104 %V 14 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007051/ %R 10.1051/cocv:2007051 %G en %F COCV_2008__14_1_71_0

Kałamajska, Agnieszka; Kružík, Martin. Oscillations and concentrations in sequences of gradients. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 71-104. doi : 10.1051/cocv:2007051. http://www.numdam.org/articles/10.1051/cocv:2007051/

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

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

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

,[4] 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] 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] ${W}^{1,p}$-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl

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

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

,[9] Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108 (1987) 667-689. | MR | Zbl

and ,[10] Linear Operators, Part I. Interscience, New York (1967). | Zbl

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

,[12] Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992). | MR | Zbl

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

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

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

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

,[17] Homogenization of periodic nonconvex integral functionals in terms of Young measures. ESAIM: COCV 12 (2006) 35-51. | Numdam | MR | Zbl

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

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

,[20] On weak lower semicontinuity of integral functionals along concentrating sequences (in preparation).

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

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

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

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

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

and ,[26] Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim. 20 (1999) 511-530. | Zbl

and ,[27] 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

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

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

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

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

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

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

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

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

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

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

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

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

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

,[43] 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

,*Cited by Sources: *