Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stability of weakly almost conformal mappings

Authors: Baisheng Yan and Zhengfang Zhou
Journal: Proc. Amer. Math. Soc. 126 (1998), 481-489
MSC (1991): Primary 49J10, 35J50, 30C62
MathSciNet review: 1415344
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

  • 1. Emilio Acerbi and Nicola Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), no. 2, 125–145. MR 751305, 10.1007/BF00275731
  • 2. John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 0475169
  • 3. J. M. Ball and F. Murat, 𝑊^{1,𝑝}-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), no. 3, 225–253. MR 759098, 10.1016/0022-1236(84)90041-7
  • 4. Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
  • 5. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 0346619
  • 6. De Figueiredo, D. G., ``The Ekeland Variational Principle with Applications and Detours," Tata Institute Lecture, Springer-Verlag, Berlin, Heidelberg, New York, 1989.
  • 7. F. W. Gehring, The 𝐿^{𝑝}-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 0402038
  • 8. Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • 9. Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
  • 10. L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 1, 17–35 (English, with English and French summaries). MR 1259100
  • 11. Tadeusz Iwaniec, 𝑝-harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136 (1992), no. 3, 589–624. MR 1189867, 10.2307/2946602
  • 12. Tadeusz Iwaniec and Gaven Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), no. 1, 29–81. MR 1208562, 10.1007/BF02392454
  • 13. T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143–161. MR 1288682, 10.1515/crll.1994.454.143
  • 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
  • 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. Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644
  • 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. 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

Similar Articles

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

Retrieve articles in all journals with MSC (1991): 49J10, 35J50, 30C62

Additional Information

Baisheng Yan
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Zhengfang Zhou
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Received by editor(s): February 26, 1996
Received by editor(s) in revised form: August 12, 1996
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society