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The limit of \(W^{1,1}\) homeomorphisms with finite distortion. (English) Zbl 1157.30016

Authors’ summary: We show that the limit \(f\) of a weakly convergent sequence of \(W ^{1,1}\) homeomorphisms \(f_j\) with finite distortion has finite distortion as well, provided that it is a homeomorphism. Moreover, the lower semicontinuity of the distortions is deduced both in case of outer and inner distortion.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26B10 Implicit function theorems, Jacobians, transformations with several variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984) · Zbl 0565.49010 · doi:10.1007/BF00275731
[2] Ambrosio L., Fusco N., Pallara D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000) · Zbl 0957.49001
[3] Astala K., Iwaniec T., Martin G., Onninen J.: Extremal mappings of finite distortion. Proc. Lond. Math. Soc. 91(3), 655–702 (2005) · Zbl 1089.30013 · doi:10.1112/S0024611505015376
[4] Brooks J.K., Chacon R.V.: Continuity and compactness of measures. Adv. Math. 37, 16–26 (1980) · Zbl 0463.28003 · doi:10.1016/0001-8708(80)90023-7
[5] Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space W 1,n-1 (2007) (preprint) · Zbl 1210.46023
[6] Dacorogna B., Marcellini P.: Semicontinuité pour des integrands polyconvexes sans continuité des determinants. C .R. Acad. Sci. Paris Sér. I Math. 311, 393–395 (1990) · Zbl 0723.49007
[7] Dal Maso G., Sbordone C.: Weak lower semicontinuity of polyconvex integrals: a borderline case. Math. Z. 218, 603–609 (1995) · Zbl 0822.49010 · doi:10.1007/BF02571927
[8] Fusco N., Hutchinson J.: A direct proof for lower semicontinuity of polyconvex functionals. Manusc. Math. 87, 35–50 (1995) · Zbl 0874.49015 · doi:10.1007/BF02570460
[9] Federer H.: Geometric measure theory. Springer, Heidelberg (1969) · Zbl 0176.00801
[10] Gehring F., Iwaniec T.: The limit of mappings with finite distortion. Ann. Acad. Sci. Fenn. A I 24, 253–264 (1999) · Zbl 0929.30016
[11] Greco, L., Sbordone, C., Trombetti, C.: A note on planar homeomorphisms (2007) (preprint) · Zbl 1210.30007
[12] Hencl S., Koskela P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180, 75–95 (2006) · Zbl 1151.30325 · doi:10.1007/s00205-005-0394-1
[13] Hencl S., Koskela P., Malý J.: Regularity of the inverse of a Sobolev homeomorphism in space. Proc. R. Soc. Edinb. A 36, 1267–1285 (2006) · Zbl 1122.30015 · doi:10.1017/S0308210500004972
[14] Hencl S., Koskela P., Onninen J.: A note on extremal mappings of finite distortion. Math. Res. Lett. 12, 231–238 (2005) · Zbl 1079.30024
[15] Iwaniec T., Martin G.: Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs. Clarendon Press, Oxford (2001) · Zbl 1045.30011
[16] Malý, J.: Lectures on change of variables in integrals, preprint 305, Department of Mathematics University of Helsinki
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