## A sharp version of Zhang’s theorem on truncating sequences of gradients

HTML articles powered by AMS MathViewer

- by Stefan Müller PDF
- Trans. Amer. Math. Soc.
**351**(1999), 4585-4597 Request permission

## 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 $\| {\operatorname {dist}} (Dv_j, K)\|_\infty \to 0$ 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 $\|Dv_j \|_\infty \le C$. 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

- Emilio Acerbi and Nicola Fusco,
*Semicontinuity problems in the calculus of variations*, Arch. Rational Mech. Anal.**86**(1984), no. 2, 125–145. MR**751305**, DOI 10.1007/BF00275731 - Emilio Acerbi and Nicola Fusco,
*An approximation lemma for $W^{1,p}$ functions*, Material instabilities in continuum mechanics (Edinburgh, 1985–1986) Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 1–5. MR**970512** - John M. Ball,
*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rational Mech. Anal.**63**(1976/77), no. 4, 337–403. MR**475169**, DOI 10.1007/BF00279992 - Bernard Dacorogna,
*Direct methods in the calculus of variations*, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR**990890**, DOI 10.1007/978-3-642-51440-1 - Ivar Ekeland and Roger Temam,
*Analyse convexe et problèmes variationnels*, Collection Études Mathématiques, Dunod, Paris; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). MR**0463993** - Lawrence C. Evans,
*Weak convergence methods for nonlinear partial differential equations*, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1034481**, DOI 10.1090/cbms/074 - I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients,
*SIAM J. Math. Anal.***29**(1998), 736–756. - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - David Kinderlehrer and Pablo Pedregal,
*Characterizations of Young measures generated by gradients*, Arch. Rational Mech. Anal.**115**(1991), no. 4, 329–365. MR**1120852**, DOI 10.1007/BF00375279 - David Kinderlehrer and Pablo Pedregal,
*Gradient Young measures generated by sequences in Sobolev spaces*, J. Geom. Anal.**4**(1994), no. 1, 59–90. MR**1274138**, DOI 10.1007/BF02921593 - J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, Ph.D. Thesis, Technical University of Denmark, Lyngby.
- J. Kristensen, On the non-locality of quasiconvexity,
*Ann. Inst. H. Poincaré Anal. Non Linéaire***16**(1999), 1–13. - Fon Che Liu,
*A Luzin type property of Sobolev functions*, Indiana Univ. Math. J.**26**(1977), no. 4, 645–651. MR**450488**, DOI 10.1512/iumj.1977.26.26051 - P. Hebroni,
*Sur les inverses des éléments dérivables dans un anneau abstrait*, C. R. Acad. Sci. Paris**209**(1939), 285–287 (French). MR**14** - Charles B. Morrey Jr.,
*Multiple integrals in the calculus of variations*, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR**0202511** - Pablo Pedregal,
*Parametrized measures and variational principles*, Progress in Nonlinear Differential Equations and their Applications, vol. 30, Birkhäuser Verlag, Basel, 1997. MR**1452107**, DOI 10.1007/978-3-0348-8886-8 - M. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, to appear in
*Ann. Inst. H. Poincaré Anal. Non Linéaire*. - Rudolph E. Langer,
*The boundary problem of an ordinary linear differential system in the complex domain*, Trans. Amer. Math. Soc.**46**(1939), 151–190 and Correction, 467 (1939). MR**84**, DOI 10.1090/S0002-9947-1939-0000084-7 - Richard Schoen and Karen Uhlenbeck,
*Boundary regularity and the Dirichlet problem for harmonic maps*, J. Differential Geom.**18**(1983), no. 2, 253–268. MR**710054** - Vladimír Šverák,
*Lower-semicontinuity of variational integrals and compensated compactness*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1153–1158. MR**1404015** - Kewei Zhang,
*A construction of quasiconvex functions with linear growth at infinity*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**19**(1992), no. 3, 313–326. MR**1205403**

## Additional Information

**Stefan Müller**- Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
- Email: sm@mis.mpg.de
- Received by editor(s): June 23, 1997
- Published electronically: July 21, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 4585-4597 - MSC (1991): Primary 49J45
- DOI: https://doi.org/10.1090/S0002-9947-99-02520-9
- MathSciNet review: 1675222