Density not realizable as the Jacobian determinant of a bilipschitz map

• Autorzy: Kaluza V.

Identyfikatory

DOI

10.1515/jaa-2016-0004

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

jacobian; bi-Lipschitz; nonrealizable; separated net

PL

jakobian; bi-Lipschitz; sieć wydzielona

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 1
• Strony: 37--47
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Kaluza V.

• 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.