Proceedings of the American Mathematical Society

Author(s): Baisheng Yan; Zhengfang Zhou
Journal: Proc. Amer. Math. Soc. 126 (1998), 481-489.
MSC (1991): Primary 49J10, 35J50, 30C62
Retrieve article in: PDF
This article is available free of charge

Abstract: We prove a stability of weakly almost conformal mappings in $W^{1, p}(\Omega;\mathbf {R}^n)$ for $p$

not too far below the dimension $ n$

by studying the $W^{1, p}$ -quasiconvex hull of the set $\mathcal C_n$ of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle.

1.: Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125-145. MR 85m:49021
2.: Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977), 337-403. MR 57:14788
3.: Ball, J. M. and Murat, F., $W^{1,p}$ -Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225-253. MR 87g:49011a
4.: Dacorogna, B., ``Direct Methods in the Calculus of Variations," Springer-Verlag, Berlin, Heidelberg, New York, 1989. MR 90e:49001
5.: Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. MR 49:11344
6.: De Figueiredo, D. G., ``The Ekeland Variational Principle with Applications and Detours," Tata Institute Lecture, Springer-Verlag, Berlin, Heidelberg, New York, 1989.
7.: Gehring, F. W., The -integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. MR 53:5861
8.: Giaquinta, M., ``Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Princeton University Press, Princeton, 1983. MR 86b:49003
9.: Giaquinta, M., ``Introduction to Regularity Theory for Nonlinear Elliptic Systems," Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, Berlin, 1993. MR 94g:49002
10.: Greco, L. and Iwaniec, T., New inequalities for the Jacobian, Ann. Inst. H. Poincaré, Analyse non linéaire, 11(1) (1994), 17-35. MR 95b:42020
11.: Iwaniec, T., p-Harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624. MR 94d:30034
12.: Iwaniec, T. and Martin, G., Quasiregular mappings in even dimensions, Acta Math., 170 (1993), 29-81. MR 94m:30046
13.: Iwaniec, T. and Sbordone, C., Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. MR 95d:49035
14.: Morrey, C. B. Jr., ``Multiple Integrals in the Calculus of Variations," Springer-Verlag, Berlin, Heidelberg, New York, 1966. MR 34:2380
15.: Müller, S., \v{S}verák, V. and Yan, B., Sharp stability results for almost conformal maps in even dimensions, submitted to Journ. Geom. Analysis.
16.: Reshetnyak, Yu. G., ``Space Mappings with Bounded Distortion," Transl. Math. Mono., A.M.S., Vol. 73, 1989. MR 90d:30067
17.: \v{S}verák, V., Lower semicontinuity for variational integral functionals and compensated compactness, Proceedings of I.C.M., Zürich, 1994.
18.: Yan, B., Remarks about $W^{1,p}$ -stability of the conformal set in higher dimensions, Ann. Inst. H. Poincaré, Analyse non linéaire, 13 (6), 1996.
19.: Yan, B., On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinb., Ser. A. (to appear)
20.: Yan, B., On $W^{1,p}$ -quasiconvex hull of sets of matrices and weak convergence in Sobolev spaces, preprint, 1995.
21.: Zhang, K., A construction of quasiconvex functions with linear growth at infinity, Ann. Scuola Norm. Sup. Pisa, 19(3) (1992), 313-326. MR 94d:49018

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 49J10, 35J50, 30C62

Baisheng Yan
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: yan@math.msu.edu

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS