Let . We construct a Hölder continuous mapping of a square into such that the distributional Jacobian equals to one-dimensional Hausdorff measure on a line segment.
@article{AIHPC_2014__31_5_947_0, author = {Hencl, Stanislav and Liu, Zhuomin and Mal\'y, Jan}, title = {Distributional {Jacobian} equal to $ {\mathcal{H}}^{1}$ measure}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {947--955}, publisher = {Elsevier}, volume = {31}, number = {5}, year = {2014}, doi = {10.1016/j.anihpc.2013.08.002}, mrnumber = {3258361}, zbl = {06349274}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.002/} }
TY - JOUR AU - Hencl, Stanislav AU - Liu, Zhuomin AU - Malý, Jan TI - Distributional Jacobian equal to $ {\mathcal{H}}^{1}$ measure JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 947 EP - 955 VL - 31 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.002/ DO - 10.1016/j.anihpc.2013.08.002 LA - en ID - AIHPC_2014__31_5_947_0 ER -
%0 Journal Article %A Hencl, Stanislav %A Liu, Zhuomin %A Malý, Jan %T Distributional Jacobian equal to $ {\mathcal{H}}^{1}$ measure %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 947-955 %V 31 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.002/ %R 10.1016/j.anihpc.2013.08.002 %G en %F AIHPC_2014__31_5_947_0
Hencl, Stanislav; Liu, Zhuomin; Malý, Jan. Distributional Jacobian equal to $ {\mathcal{H}}^{1}$ measure. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 947-955. doi : 10.1016/j.anihpc.2013.08.002. http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.002/
[1] Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 no. 4 (1976/1977), 337 -403 | MR | Zbl
,[2] The Jacobian determinant revisited, Invent. Math. 185 no. 1 (2011), 17 -54 | MR | Zbl
, ,[3] Some remarks on the distributional Jacobian, Nonlinear Anal. 53 no. 7–8 (2003), 1101 -1114 | MR | Zbl
,[4] An extension of the identity , C. R. Math. Acad. Sci. Paris 348 no. 17–18 (2010), 973 -976 | MR | Zbl
, ,[5] A remark on the equality , Differential Integral Equations 6 no. 5 (1993), 1089 -1100 | MR | Zbl
,[6] Sobolev homeomorphism with zero Jacobian almost everywhere, J. Math. Pures Appl. (9) 95 no. 4 (2011), 444 -458 | MR | Zbl
,[7] Lectures on mappings of finite distortion, Lecture Notes in Mathematics , Springer (2013) | MR
, ,[8] Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs , The Clarendon Press, Oxford University Press, New York (2001) | MR
, ,[9] On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 no. 2 (1992), 129 -143 | MR | Zbl
, ,[10] Functions of bounded higher variation, Indiana Univ. Math. J. 51 no. 3 (2002), 645 -677 | MR | Zbl
, ,[11] Minimal assumptions for the integrability of the Jacobian, Ricerche Mat. 51 no. 2 (2003), 297 -311 | MR | Zbl
, ,[12] Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften vol. 130 , Springer-Verlag, New York (1966) | MR | Zbl
,[13] . A remark on the distributional determinant, C. R. Acad. Sci. Paris Sér. I Math. 311 no. 1 (1990), 13 -17 | MR | Zbl
,[14] On the singular support of the distributional determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 no. 6 (1993), 657 -696 | EuDML | Numdam | MR | Zbl
,[15] On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 no. 2 (1994), 217 -243 | EuDML | Numdam | MR | Zbl
, , ,[16] An example of an homeomorphism that is not absolutely continuous in the sense of Banach, Dokl. Akad. Nauk SSSR 201 (1971), 1053 -1054 | MR | Zbl
,[17] The weak convergence of completely additive vector-valued set functions, Sibirsk. Mat. Z̆. 9 (1968), 1386 -1394 | MR | Zbl
,[18] Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal. 100 no. 2 (1988), 105 -127 | MR | Zbl
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