Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

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
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
  • 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