Abstract
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.
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References
Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984)
Ambrosio L., Fusco N., Pallara D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000)
Astala K., Iwaniec T., Martin G., Onninen J.: Extremal mappings of finite distortion. Proc. Lond. Math. Soc. 91(3), 655–702 (2005)
Brooks J.K., Chacon R.V.: Continuity and compactness of measures. Adv. Math. 37, 16–26 (1980)
Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space W 1,n-1 (2007) (preprint)
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)
Dal Maso G., Sbordone C.: Weak lower semicontinuity of polyconvex integrals: a borderline case. Math. Z. 218, 603–609 (1995)
Fusco N., Hutchinson J.: A direct proof for lower semicontinuity of polyconvex functionals. Manusc. Math. 87, 35–50 (1995)
Federer H.: Geometric measure theory. Springer, Heidelberg (1969)
Gehring F., Iwaniec T.: The limit of mappings with finite distortion. Ann. Acad. Sci. Fenn. A I 24, 253–264 (1999)
Greco, L., Sbordone, C., Trombetti, C.: A note on planar homeomorphisms (2007) (preprint)
Hencl S., Koskela P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180, 75–95 (2006)
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)
Hencl S., Koskela P., Onninen J.: A note on extremal mappings of finite distortion. Math. Res. Lett. 12, 231–238 (2005)
Iwaniec T., Martin G.: Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs. Clarendon Press, Oxford (2001)
Malý, J.: Lectures on change of variables in integrals, preprint 305, Department of Mathematics University of Helsinki
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Fusco, N., Moscariello, G. & Sbordone, C. The limit of W 1,1 homeomorphisms with finite distortion. Calc. Var. 33, 377–390 (2008). https://doi.org/10.1007/s00526-008-0169-2
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DOI: https://doi.org/10.1007/s00526-008-0169-2