Lower semicontinuity via \(W^{1,q}\)-quasiconvexity. (English) Zbl 1277.49016
Summary: We isolate a general condition, that we call “localization principle”, on the integrand \(L:\mathbb M\to [0,\infty]\), assumed to be continuous, under which \(W^{1,q}\)-quasiconvexity with \(q\in [1,\infty]\) is a sufficient condition for \(I(u)=\int_\Omega L(\nabla u(x))\,dx\) to be sequentially weakly lower semicontinuous on \(W^{1,p}(\Omega;\mathbb R^m)\) with \(p\in ]1,\infty [\). Some applications are given.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49J10 | Existence theories for free problems in two or more independent variables |
Keywords:
weak lower semicontinuity; \(W^{1,q}\)-quasiconvexity; Young measures; equi-integrability; localization principleReferences:
| [1] | Acerbi, Emilio; Fusco, Nicola, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal., 86, 2, 125-145 (1984) · Zbl 0565.49010 |
| [2] | Balder, E. J., A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22, 4, 570-598 (1984) · Zbl 0549.49005 |
| [3] | Ball, J. M., A version of the fundamental theorem for Young measures, (PDEs and Continuum Models of Phase Transitions. PDEs and Continuum Models of Phase Transitions, Nice, 1988. PDEs and Continuum Models of Phase Transitions. PDEs and Continuum Models of Phase Transitions, Nice, 1988, Lecture Notes in Phys., vol. 344 (1989), Springer: Springer Berlin), 207-215 · Zbl 0991.49500 |
| [4] | Ball, J. M.; Murat, F., \(W^{1, p}\)-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58, 3, 225-253 (1984) · Zbl 0549.46019 |
| [5] | Ball, J. M.; Zhang, K.-W., Lower semicontinuity of multiple integrals and the biting lemma, Proc. Roy. Soc. Edinburgh Sect. A, 114, 3-4, 367-379 (1990) · Zbl 0716.49011 |
| [6] | Fonseca, Irene; Leoni, Giovanni, Modern Methods in the Calculus of Variations: \(L^p\) Spaces, Springer Monogr. Math. (2007), Springer: Springer New York · Zbl 1153.49001 |
| [7] | Fonseca, Irene; Müller, Stefan; Pedregal, Pablo, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal., 29, 3, 736-756 (1998), (electronic) · Zbl 0920.49009 |
| [8] | Kinderlehrer, David; Pedregal, Pablo, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal., 23, 1, 1-19 (1992) · Zbl 0757.49014 |
| [9] | Kinderlehrer, David; Pedregal, Pablo, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4, 1, 59-90 (1994) · Zbl 0808.46046 |
| [11] | Kristensen, Jan, Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand, Proc. Roy. Soc. Edinburgh Sect. A, 127, 4, 797-817 (1997) · Zbl 0887.49011 |
| [12] | Kristensen, Jan, Lower semicontinuity in spaces of weakly differentiable functions, Math. Ann., 313, 4, 653-710 (1999) · Zbl 0924.49012 |
| [13] | Morrey, Charles B., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2, 25-53 (1952) · Zbl 0046.10803 |
| [14] | Pedregal, Pablo, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl., vol. 30 (1997), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0879.49017 |
| [15] | Pedregal, Pablo, Variational Methods in Nonlinear Elasticity (2000), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0941.74002 |
| [16] | Sychev, M. A., A new approach to Young measure theory, relaxation and convergence in energy, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 6, 773-812 (1999) · Zbl 0943.49012 |
| [17] | Sychev, M. A., Characterization of homogeneous gradient Young measures in case of arbitrary integrands, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 29, 3, 531-548 (2000) · Zbl 1067.49009 |
| [18] | Sychev, M. A., Young measures as measurable functions and their applications to variational problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts., 35, 34, 228-229 (2004) · Zbl 1099.49013 |
| [19] | Sychev, M. A., Semicontinuity and relaxation theorems for integrands satisfying the fast growth condition, Sibirsk. Mat. Zh., 46, 3, 679-697 (2005) · Zbl 1095.49014 |
| [20] | Young, L. C., Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. Lett. Vars., 30, 212-234 (1937) · Zbl 0019.21901 |
| [21] | Ziemer, William P., Weakly differentiable functions, (Sobolev Spaces and Functions of Bounded Variation. Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math., vol. 120 (1989), Springer-Verlag: Springer-Verlag New York) · Zbl 0692.46022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
