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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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Stability of weakly almost conformal mappings
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by Baisheng Yan and Zhengfang Zhou
Proc. Amer. Math. Soc. 126 (1998), 481-489
DOI: https://doi.org/10.1090/S0002-9939-98-04079-9
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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
  • 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
  • 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
  • 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, DOI 10.1016/0022-1236(84)90041-7
  • 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
  • I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
  • De Figueiredo, D. G., “The Ekeland Variational Principle with Applications and Detours," Tata Institute Lecture, Springer-Verlag, Berlin, Heidelberg, New York, 1989.
  • F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
  • 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
  • Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
  • L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 1, 17–35 (English, with English and French summaries). MR 1259100, DOI 10.1016/S0294-1449(16)30194-9
  • Tadeusz Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136 (1992), no. 3, 589–624. MR 1189867, DOI 10.2307/2946602
  • Tadeusz Iwaniec and Gaven Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), no. 1, 29–81. MR 1208562, DOI 10.1007/BF02392454
  • T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143–161. MR 1288682, DOI 10.1515/crll.1994.454.143
  • 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, DOI 10.1007/978-3-540-69952-1
  • Müller, S., Šverák, V. and Yan, B., Sharp stability results for almost conformal maps in even dimensions, submitted to Journ. Geom. Analysis.
  • 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, DOI 10.1090/mmono/073
  • Šverák, V., Lower semicontinuity for variational integral functionals and compensated compactness, Proceedings of I.C.M., Zürich, 1994.
  • 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.
  • Yan, B., On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinb., Ser. A. (to appear)
  • Yan, B., On $W^{1,p}$-quasiconvex hull of sets of matrices and weak convergence in Sobolev spaces, preprint, 1995.
  • 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
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Bibliographic Information
  • Baisheng Yan
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
  • MR Author ID: 348214
  • Email: yan@math.msu.edu
  • Zhengfang Zhou
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
  • Email: zfzhou@math.msu.edu
  • Received by editor(s): February 26, 1996
  • Received by editor(s) in revised form: August 12, 1996
  • Communicated by: Albert Baernstein II
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 481-489
  • MSC (1991): Primary 49J10, 35J50, 30C62
  • DOI: https://doi.org/10.1090/S0002-9939-98-04079-9
  • MathSciNet review: 1415344