Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:20:20.535Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasiconvexity at the boundary and concentration effectsgenerated by gradients∗

Published online by Cambridge University Press:  17 May 2013

Martin Kružík
Martin Kružík*
Affiliation:
Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Praha 6, Czech Republic. kruzik@utia.cas.cz
Get access

Abstract

We characterize generalized Young measures, the so-called DiPerna–Majda measures whichare generated by sequences of gradients. In particular, we precisely describe thesemeasures 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 onweak W1,2(Ω; ℝ3) sequentialcontinuity ofu → a· [Cof∇u] ϱ,where Ω ⊂ ℝ3 has a smooth boundary and a,ϱare certain smooth mappings.

Keywords

Bounded sequences of gradientsconcentrationsoscillationsquasiconvexity at the boundaryweak lower semicontinuity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 679 - 700
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alibert, J.J. and Bouchitté, G., Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997) 125–145. Google Scholar
J.M. Ball, 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.
Ball, J.M. and Marsden, J., Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86 (1984) 251–277. Google Scholar
Ball, J.M. and Murat, F., W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. Google Scholar
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer, Berlin (1989).
Di Perna, R.J. and Majda, A.J., Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667–689. Google Scholar
N. Dunford and J.T. Schwartz, Linear Operators, Part I, Interscience, New York (1967).
R. Engelking, General topology, 2nd edition. PWN, Warszawa (1985).
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Inc. Boca Raton (1992).
Fonseca, I., Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh 120A (1992) 95–115. Google Scholar
Fonseca, I. and Kruížk, M., Oscillations and concentrations generated by 𝒜-free mappings and weak lower semicontinuity of integral functionals. ESAIM: COCV 16 (2010) 472–502. Google Scholar
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. Google Scholar
Grabovskya, Y. and Mengesha, T., Direct approach to the problem of strong local minima in calculus of variations. Calc. Var. 29 (2007) 59–83. Google Scholar
Hogan, J., Li, C., McIntosh, A. and Zhang, K., Global higher integrability of Jacobians on bounded domains. Ann. l’Inst. Henri Poincaré Sect. C 17 (2000) 193–217. Google Scholar
A., Kałamajska and M., Kruížk, Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71–104.
Kinderlehrer, D. and Pedregal, P., Characterization of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115 (1991) 329–365. Google Scholar
Kinderlehrer, D. and Pedregal, P., Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23 (1992) 1–19. Google Scholar
Kinderlehrer, D. and Pedregal, P., Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–90. Google Scholar
Kristensen, J. and Rindler, F., Characterization of generalized gradient Young measures generated by sequences in W 1,1 and BV. Arch. Ration. Mech. Anal. 197 (2010) 539–598. Google Scholar
Krömer, S., On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals. Adv. Calc. Var. 3 (2010) 378–408. Google Scholar
M., Kružík and Luskin, M., The computation of martensitic microstructure with piecewise laminates. J. Sci. Comput. 19 (2003) 293–308. Google Scholar
M., Kružík and Roubcíek, T., On the measures of DiPerna and Majda. Math. Bohemica 122 (1997) 383–399. Google Scholar
M., Kružík and Roubcíek, T., Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim. 20 (1999) 511–530. Google Scholar
Meyers, N.G., Quasi-convexity and lower semicontinuity of multiple integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125–149. Google Scholar
Mielke, A. and Sprenger, P., Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition. J. Elasticity 51 (1998) 23–41. Google Scholar
C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
Müller, S., Higher integrability of determinants and weak convergence in L 1. J. Reine Angew. Math. 412 (1990) 20–34. Google Scholar
S., Müller, Variational models for microstructure and phase transisions, Lect. Notes Math., vol. 1713. Springer, Berlin (1999) 85–210. Google Scholar
P. Pedregal, Parametrized Measures and Variational Principles. Birkäuser, Basel (1997).
T. Roubčíek, Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997).
T., Roubčíek and Kruıžk, M., Microstructure evolution model in micromagnetics. Zeit. Angew. Math. Phys. 55 (2004) 159–182. Google Scholar
T., Roubčíek and Kruıžk, M., Mesoscopical model for ferromagnets with isotropic hardening. Zeit. Angew. Math. Phys. 56 (2005) 107–135. Google Scholar
Schonbek, M.E., Convergence of solutions to nonlinear dispersive equations. Commun. Partial Differ. Equ. 7 (1982) 959–1000. Google Scholar
M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997).
M. Šilhavý, 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.
P. Sprenger, Quasikonvexität am Rande und Null-Lagrange-Funktionen in der nichtkonvexen Variationsrechnung. Ph.D. thesis, Universität Hannover (1996).
L. Tartar, 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).
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, Proc. of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, edited by N. Antonič et al. Springer, Berlin (2002).
M. Valadier, Young measures, in Methods of Nonconvex Analysis, Lect. Notes Math., vol. 1446, edited by A. Cellina. 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. C. R. de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 30 (1937) 212–234. Google Scholar