Voir la notice de l'article provenant de la source Numdam
We characterize generalized Young measures, the so-called DiPerna-Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of ℝm × n by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, Arch. Ration. Mech. Anal. 86 (1984) 251-277]. As a consequence we get new results on weak W1,2(Ω; ℝ3) sequential continuity of u → a· [Cof∇u] ϱ, where Ω ⊂ ℝ3 has a smooth boundary and a,ϱ are certain smooth mappings.
@article{COCV_2013__19_3_679_0,
author = {Kru\v{z}{\'\i}k, Martin},
title = {Quasiconvexity at the boundary and concentration effects generated by gradients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {679--700},
publisher = {EDP-Sciences},
volume = {19},
number = {3},
year = {2013},
doi = {10.1051/cocv/2012028},
mrnumber = {3092357},
zbl = {1277.49014},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012028/}
}
TY - JOUR AU - Kružík, Martin TI - Quasiconvexity at the boundary and concentration effects generated by gradients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 679 EP - 700 VL - 19 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012028/ DO - 10.1051/cocv/2012028 LA - en ID - COCV_2013__19_3_679_0 ER -
%0 Journal Article %A Kružík, Martin %T Quasiconvexity at the boundary and concentration effects generated by gradients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 679-700 %V 19 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012028/ %R 10.1051/cocv/2012028 %G en %F COCV_2013__19_3_679_0
Kružík, Martin. Quasiconvexity at the boundary and concentration effects generated by gradients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 679-700. doi : 10.1051/cocv/2012028. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012028/
[1] and , Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997) 125-145. | Zbl | MR | EuDML
[2] , A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transition. Lect. Notes Phys., vol. 344, edited by M. Rascle, D. Serre and M. Slemrod. Springer, Berlin (1989) 207-215. | Zbl | MR
[3] and , Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86 (1984) 251-277. | Zbl | MR
[4] and , W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | Zbl | MR
[5] , Direct Methods in the Calculus of Variations. Springer, Berlin (1989). | Zbl | MR
[6] and , Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667-689. | Zbl | MR
[7] and , Linear Operators, Part I, Interscience, New York (1967). | Zbl
[8] , General topology, 2nd edition. PWN, Warszawa (1985). | Zbl
[9] and , Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992). | Zbl | MR
[10] , Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh 120A (1992) 95-115. | Zbl | MR
[11] and , Oscillations and concentrations generated by 𝒜-free mappings and weak lower semicontinuity of integral functionals. ESAIM: COCV 16 (2010) 472-502. | Zbl | MR | mathdoc-id
[12] , and , Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | Zbl | MR
[13] and , Direct approach to the problem of strong local minima in calculus of variations. Calc. Var. 29 (2007) 59-83. | Zbl | MR
[14] , , and , Global higher integrability of Jacobians on bounded domains. Ann. l'Inst. Henri Poincaré Sect. C 17 (2000) 193-217. | Zbl | MR | mathdoc-id
[15] and , Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71-104. | Zbl | MR | mathdoc-id
[16] and , Characterization of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329-365. | Zbl | MR
[17] and , Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23 (1992) 1-19. | Zbl | MR
[18] and , Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | Zbl | MR
[19] and , Characterization of generalized gradient Young measures generated by sequences in W1,1 and BV. Arch. Ration. Mech. Anal. 197 (2010) 539-598. | Zbl | MR
[20] , On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals. Adv. Calc. Var. 3 (2010) 378-408. | Zbl | MR
[21] and , The computation of martensitic microstructure with piecewise laminates. J. Sci. Comput. 19 (2003) 293-308. | Zbl | MR
[22] and , On the measures of DiPerna and Majda. Math. Bohemica 122 (1997) 383-399. | Zbl
[23] and , Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim. 20 (1999) 511-530. | Zbl
[24] , Quasi-convexity and lower semicontinuity of multiple integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125-149. | Zbl | MR
[25] and , Quasiconvexity at the boundary and a simple variational formulation of Agmon's condition. J. Elasticity 51 (1998) 23-41. | Zbl | MR
[26] , Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966). | Zbl
[27] , Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math. 412 (1990) 20-34. | Zbl | MR
[28] , Variational models for microstructure and phase transisions, Lect. Notes Math., vol. 1713. Springer, Berlin (1999) 85-210. | Zbl | MR
[29] , Parametrized Measures and Variational Principles. Birkäuser, Basel (1997). | Zbl | MR
[30] , Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997). | Zbl
[31] and , Microstructure evolution model in micromagnetics. Zeit. Angew. Math. Phys. 55 (2004) 159-182. | Zbl
[32] and , Mesoscopical model for ferromagnets with isotropic hardening. Zeit. Angew. Math. Phys. 56 (2005) 107-135. | Zbl
[33] , Convergence of solutions to nonlinear dispersive equations. Commun. Partial Differ. Equ. 7 (1982) 959-1000. | Zbl | MR
[34] , The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997). | Zbl
[35] , Phase transitions with interfacial energy: Interface Null Lagrangians, Polyconvexity, and Existence, in IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries, vol. 21, edited by K. Hackl. Springer (2010) 233-244.
[36] , Quasikonvexität am Rande und Null-Lagrange-Funktionen in der nichtkonvexen Variationsrechnung. Ph.D. thesis, Universität Hannover (1996). | Zbl
[37] , Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics, Heriott-Watt Symposium IV, Pitman Res. Notes Math., vol. 39, edited by R.J. Knops. (1979). | Zbl | MR
[38] , 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, Proc. of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, edited by N. Antonič et al. Springer, Berlin (2002). | Zbl | MR
[39] , Young measures, in Methods of Nonconvex Analysis, Lect. Notes Math., vol. 1446, edited by A. Cellina. Springer, Berlin (1990) 152-188. | Zbl | MR
[40] , Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | Zbl | MR
[41] , Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 30 (1937) 212-234. | Zbl | JFM
Cité par Sources :