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Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen's construction is simpler than the Burago–Kleiner one, and we provide a full proof of its nonrealizability, which has not been available in the literature.
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Tom
Strony
37--47
Opis fizyczny
Bibliogr. 7 poz.
Twórcy
autor
- Department of Applied Mathematics, Charles University in Prague, Malostranské nám. 25, 118 00 Praha 1, Czech Republic
Bibliografia
- [1] D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct Anal. 8 (1998), 273-282.
- [2] B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincare Anal Non Linéaire 7 (1990), no. 1,1-26.
- [3] D. H. Fremlin, Measure Theory. Vol. 2: Broad Foundations in Measure Theory, Torres Fremlin, Colchester, 2000.
- [4] M. L. Gromov, Geometric Group Theory. Volume 2: Asymptotic Invariants of infinite Groups, London Math. Soc. Lecture Note Ser. 182, Cambridge University Press, Cambridge, 1993.
- [5] V. Kałuża, Lipschitz mappings of discrete sets (in Czech), Bachelor thesis, Charles University in Prague, Prague, 2012.
- [6] C. T. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8 (1998), 304-314.
- [7] D. Ye, Prescribing the Jacobian determinant in Sobolev spaces, Ann. Inst H. Poincare Anal. Non Linéaire 11 (1994), no. 3, 275-296.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
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