By Stanislav Hencl
ISBN-10: 3319031724
ISBN-13: 9783319031729
ISBN-10: 3319031732
ISBN-13: 9783319031736
In this booklet we introduce the category of mappings of finite distortion as a generalization of the category of mappings of bounded distortion. Connections with types of nonlinear elasticity also are mentioned. We examine continuity homes, habit of our mappings on null units, topological houses like openness and discreteness, regularity of the capability inverse mappings and lots of different aspects.
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X; r//. V / is open. y/ is not discrete in ˝. It follows that we can find x 2 ˝ such that each neighborhood of x contains infinitely many preimages of y. x; r//. C \ kC1 ' D 1. x/ dx Ui kC1 i D1 t u which gives us the desired contradiction. 4 we will use convolution approximations and for the approximations we will use the following technical estimate. 19. 0/ D 0. 8) ˝ Proof. Fix i 2 f1; : : : ; ng. 4). n/ > n 1 for n 3. 18. By the translation x ! x/ y it is enough to show this only for y D 0; under the assumption that f is not identically equal to zero.
B/. Then ˝. Z Jf D 0: B Proof. 10 applied to test functions ' such that ' Á 1 on B since we can extend f1 as zero outside B. B/, f k ! f in W 1;n 1 2 and Df k1 ! 1 ! 14. x0 ;R/ u D 0 and let > 0. x/ > C : Proof. x/ Ä 2jujB0 . x0 ; R/, C jBi j. x/ i D0 t u and the result follows. 15. x0 ; R//. x0 ;R/nF Proof. y/: This implies that at least one of the right-hand side terms is greater than 2C . 8) Z ÄC jDf jn 1 ı 1Cı : fjDf j> C2 g We now show that the last one of these terms is small. e C t/ fjDf j>t g 1 !
R n we can take the Tietze extension fO W R n ! R n , which is a continuous mapping. C; fO; U / and by (a) this is independent of the choice of fO. (f) The degree is stable under homotopy. Let H W U Œ0; 1 ! x/. x; t/. y; g; U /. 8 if both f and g are Lipschitz. The general case follows by approximation by Lipschitz mappings. f /Jf . 13. ˝; R n / be a continuous mapping. @U /. C / which satisfies R C ' D 1. Proof. 5) where fj denote the usual convolution approximations to f . For these convolution approximations, fj !