Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A sharp version of Zhang's theorem
on truncating sequences of gradients

Author: Stefan Müller
Journal: Trans. Amer. Math. Soc. 351 (1999), 4585-4597
MSC (1991): Primary 49J45
Published electronically: July 21, 1999
MathSciNet review: 1675222
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $K \subset \mathbf{R}^{mn}$ be a compact and convex set of $m \times n$ matrices and let $\{u_j\}$ be a sequence in $W_{\operatorname{loc}} ^{1,1}(\mathbf{R}^n;\mathbf{R}^m)$ that converges to $K$ in the mean, i.e. $\int _{\mathbf{R}^n} {\operatorname{dist}} (Du_j, K) \to 0$. I show that there exists a sequence $v_j$ of Lipschitz functions such that $\parallel\!\! {\operatorname{dist}} (Dv_j, K)\!\!\parallel _\infty \ \to 0$ and $\mathcal{L}^n (\{u_j \not= v_j\}) \to 0$. This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that one may assume $\parallel\!\!Dv_j\!\! \parallel _\infty \ \le C$. Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rational Mech. Anal. 115 (1991), 329-365) regarding the approximation of $\mathbf{R} \cup \{+\infty\}$ valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of $K$ can be replaced by quasiconvexity.

References [Enhancements On Off] (What's this?)

  • 1. E. Acerbi and N. Fusco, Semincontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1984), 125-145. MR 85m:49021
  • 2. E. Acerbi and N. Fusco, An approximation lemma for $W^{1,p}$ functions, in: Material instabilities in continuum mechanics and related mathematical problems (J.M. Ball, ed.), Oxford UP, 1988, pp. 1-5. MR 89m:46060
  • 3. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337-403. MR 57:14788
  • 4. B. Dacorogna, Direct methods in the calculus of variations, Springer, 1989. MR 90e:49001
  • 5. I. Ekeland and R. Temam, Convex analysis and variational problems, North Holland, Amsterdam, 1976. MR 57:3931b
  • 6. L.C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS no. 74, 1990, Amer. Math. Soc. MR 91a:35009
  • 7. I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 (1998), 736-756. CMP 98:11
  • 8. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer, 2nd ed., 1983. MR 86c:35035
  • 9. D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients, Arch. Rat. Mech. Anal. 115 (1991), 329-365. MR 92k:49089
  • 10. D. Kinderlehrer and P. Pedregal, Gradient Young measure generated by sequences in Sobolev spaces, J. Geom. Analysis 4 (1994), 59-90. MR 95f:49059
  • 11. J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, Ph.D. Thesis, Technical University of Denmark, Lyngby.
  • 12. J. Kristensen, On the non-locality of quasiconvexity, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 1-13.
  • 13. F.-C. Liu, A Luzin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645-651. MR 56:8782
  • 14. C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25-53. MR 14:992a
  • 15. C.B. Morrey, Multiple integrals in the calculus of variations, Springer, 1966. MR 34:2380
  • 16. P. Pedregal, Parametrized measures and variational principles, Birkhäuser, 1997. MR 98e:49001
  • 17. M. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.
  • 18. R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differ. Geom. 17 (1982), 307-335; 18 (1983), 329. MR 84b:58037
  • 19. R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom. 18 (1983), 253-268. MR 85b:58037
  • 20. V. \v{S}verák, Lower semicontinuity of variational integrals and compensated compactness, in: Proc. ICM 1994, vol. 2, Birkhäuser, 1995, pp. 1153-1158. MR 97h:49021
  • 21. K. Zhang, A construction of quasiconvex functions with linear growth at infinity, Ann. Scuola Norm. Sup. Pisa 19 (1992), 313-326. MR 94d:49018

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 49J45

Retrieve articles in all journals with MSC (1991): 49J45

Additional Information

Stefan Müller
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

Keywords: Young measures, quasiconvexity, truncation
Received by editor(s): June 23, 1997
Published electronically: July 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society