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Luzin’s condition (N) and Sobolev mappings. (English) Zbl 1257.26010

Summary: For the purpose of change of variables in integral, it is important to know how to verify Luzin’s condition (N) for Sobolev mappings. We survey some results on this topic sorted according to the method. We discuss the method of absolute continuity, results obtained via degree, and results based on the interplay between integrability and modulus of continuity.

MSC:

26B15 Integration of real functions of several variables: length, area, volume
28A75 Length, area, volume, other geometric measure theory
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
51M25 Length, area and volume in real or complex geometry
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[1] G. Alberti - L. Ambrosio, A geometrical approach to monotone functions in Rn, Math. Z., 230(2):259-316, 1999. · Zbl 0934.49025
[2] H. Brezis - H.-M. Nguyen, The Jacobian determinant revisited, Invent. Math., 185(1):17-54, 2011. 463 · Zbl 1230.46029 · doi:10.1007/s00222-010-0300-9
[3] L. Cesari, Sulle trasformazioni continue, Ann. Mat. Pura Appl. (4), 21:157-188, 1942. · Zbl 0028.21004 · doi:10.1007/BF02412407
[4] A. Cianchi - L. Pick, Sobolev embeddings into BMO, VMO, and Ll, Ark. Mat., 36(2):317-340, 1998.
[5] M. Cso\"rnyei, Absolutely continuous functions of Rado, Reichelderfer, and Maly\', J. Math. Anal. Appl., 252(1):147-166, 2000.
[6] C. De Lellis, Some remarks on the distributional Jacobian, Nonlinear Anal., 53(7-8):1101-1114, 2003. · Zbl 1025.49030
[7] C. De Lellis - F. Ghiraldin, An extension of the identity Det \frac{1}{4} det, C. R. Math. Acad. Sci. Paris, 348(17-18):973-976, 2010.
[8] L. C. Evans - R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[9] H. Federer, Surface area. II, Trans. Amer. Math. Soc., 55:438-456, 1944. · Zbl 0060.14003
[10] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissen- schaften, Band 153. Springer-Verlag New York Inc., New York, 1969 (Second edition 1996).
[11] I. Fonseca - W. Gangbo, Degree theory in analysis and applications, volume 2 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1995. Oxford Science Publications. · Zbl 0852.47030
[12] F. Giannetti - T. Iwaniec - J. Onninen - A. Verde, Estimates of Jacobians by subdeterminants, J. Geom. Anal., 12(2):223-254, 2002. · Zbl 1053.42024 · doi:10.1007/BF02922041
[13] M. Giaquinta - G. Modica - J. Souc\check ek, Area and the area formula, In Proceed- ings of the Second International Conference on Partial Di\?erential Equations (Italian) (Milan, 1992), volume 62, pages 53-87 (1994), 1992. · Zbl 0828.49022
[14] M. Giaquinta - G. Modica - J. Souc\check ek, Cartesian currents in the calculus of varia- tions. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1998. Cartesian currents.
[15] V. M. Gol’dshtei\?n - Y. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, volume 54 of Mathematics and its Applications (Soviet Series), Kluwer Aca- demic Publishers Group, Dordrecht, 1990. Translated and revised from the 1983 Russian original, Translated by O. Korneeva.
[16] V. M. Gol’dstein - S. Vodop’yanov, Quasiconformal mappings and spaces of functions with generalized first derivatives, Siberian Math. J., 17(3):515-531, 1977.
[17] L. Greco, A remark on the equality det Df \frac{1}{4} Det Df , Di\?erential Integral Equations, 6(5):1089-1100, 1993. · Zbl 0784.49013
[18] L. Greco, Sharp integrability of nonnegative Jacobians, Rend. Mat. Appl. (7), 18(3):585-600, 1998. · Zbl 0991.26005
[19] L. Greco - T. Iwaniec - G. Moscariello, Limits of the improved integrability of the volume forms, Indiana Univ. Math. J., 44(2):305-339, 1995. · Zbl 0855.42009 · doi:10.1512/iumj.1995.44.1990
[20] P. HajŁasz, Change of variables formula under minimal assumptions, Colloq. Math., 64(1):93-101, 1993. · Zbl 0840.26009
[21] P. HajŁasz, Sobolev mappings, co-area formula and related topics, In Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), pages 227- 254. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000. · Zbl 0988.28002
[22] P. HajŁasz - J. T. Tyson, Sobolev Peano cubes, Michigan Math. J., 56(3):687-702, 2008. p. ko ske la, j. ma ly ánd t. zu \" r cher · Zbl 1165.28004 · doi:10.1307/mmj/1231770368
[23] C. Hamburger, Some properties of the degree for a class of Sobolev maps, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455(1986):2331-2349, 1999. · Zbl 0955.46020 · doi:10.1098/rspa.1999.0406
[24] T. Iwaniec, On the concept of the weak Jacobian and Hessian, In Papers on analysis, volume 83 of Rep. Univ. Jyva\"skyla\" Dep. Math. Stat., pages 181-205. Univ. Jyva\"skyla\", Jyva\"skyla\", 2001. · Zbl 1007.35015
[25] T. Iwaniec - C. Sbordone, On the integrability of the Jacobian under minimal hypoth- eses, Arch. Rational Mech. Anal., 119(2):129-143, 1992. · Zbl 0766.46016
[26] R. L. Jerrard - H. M. Soner, Functions of bounded higher variation, Indiana Univ. Math. J., 51(3):645-677, 2002. · Zbl 1057.49036 · doi:10.1512/iumj.2002.51.2229
[27] P. W. Jones - N. G. Makarov, Density properties of harmonic measure, Ann. of Math. (2), 142(3):427-455, 1995. · Zbl 0842.31001 · doi:10.2307/2118551
[28] J. Kauhanen, Failure of the condition N below W 1; n, Ann. Acad. Sci. Fenn. Math., 27(1):141-150, 2002. · Zbl 1027.26015
[29] J. Kauhanen - P. Koskela - J. Maly\', On functions with derivatives in a Lorentz space, Manuscripta Math., 100(1):87-101, 1999. · Zbl 0976.26004
[30] J. Kauhanen - P. Koskela - J. Maly\', Mappings of finite distortion: condition N, Michigan Math. J., 49(1):169-181, 2001.
[31] J. Kauhanen - P. Koskela - J. Maly\' - J. Onninen - X. Zhong, Mappings of finite distortion: sharp Orlicz-conditions, Rev. Mat. Iberoamericana, 19(3):857-872, 2003. · Zbl 1059.30017
[32] A. Kauranen - P. Koskela - A. Zapadinskaya, Planar Sobolev mappings and area, In preparation, 2011.
[33] P. Koskela - J. Maly\' - T. Zu\"rcher, Lusin’s condition \eth NÞ and modulus of continu- ity, In preparation, 2011.
[34] P. Koskela - X. Zhong, Minimal assumptions for the integrability of the Jacobian, Ricerche Mat., 51(2):297-311 (2003), 2002. · Zbl 1096.26005
[35] J. Maly\', Absolutely continuous functions of several variables, J. Math. Anal. Appl., 231(2):492-508, 1999. · Zbl 0924.26008
[36] J. Maly\', Su\promille cient conditions for change of variables in integral, In Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), pages 370- 386. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000.
[37] J. Maly\' - O. Martio, Lusin’s condition \eth NÞ and mappings of the class W 1; n, J. Reine Angew. Math., 458:19-36, 1995.
[38] M. Marcus - V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc., 79:790-795, 1973. · Zbl 0275.49041 · doi:10.1090/S0002-9904-1973-13319-1
[39] O. Martio - W. P. Ziemer, Lusin’s condition \eth NÞ and mappings with nonnegative Jacobians, Michigan Math. J., 39(3):495-508, 1992. · Zbl 0807.46032
[40] S. Mu\"ller, Det \frac{1}{4} det. A remark on the distributional determinant, C. R. Acad. Sci. Paris Seŕ. I Math., 311(1):13-17, 1990.
[41] S. Mu\"ller - T. Qi - B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. PoincareÁnal. Non Lineáire, 11(2):217-243, 1994. · Zbl 0863.49002
[42] S. Mu\"ller - S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal., 131(1):1-66, 1995. · Zbl 0836.73025
[43] T. Rado - P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaf- ten in Einzeldarstellungen mit besonderer Beru\"cksichtigung der Anwendungsgebiete, Bd. LXXV. Springer-Verlag, Berlin, 1955. 465 · Zbl 0067.03506
[44] J. G. Res\check etnjak, Certain geometric properties of functions and mappings with general- ized derivatives, Sibirsk. Mat. Z \? ., 7:886-919, 1966.
[45] J. G. Res\check etnjak, Spatial mappings with bounded distortion, Sibirsk. Mat. Z \? ., 8:629- 658, 1967.
[46] Y. G. Reshetnyak, The N condition for spatial mappings of the class W 1 , Sibirsk. n; loc Mat. Zh., 28(5):149-153, 1987. · Zbl 0634.30023
[47] Y. G. Reshetnyak, Space mappings with bounded distortion, volume 73 of Transla- tions of Mathematical Monographs. American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. · Zbl 0667.30018
[48] E. M. Stein, Editor’s note: the di\?erentiability of functions in Rn, Ann. of Math. (2), 113(2):383-385, 1981.
[49] V. S\check veraḱ, Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal., 100(2):105-127, 1988.
[50] D. Swanson, Area, coarea, and approximation in W 1; 1, Ark. Mat., 45(2):381-399, 2007. · Zbl 1163.46302 · doi:10.1007/s11512-007-0051-z
[51] J. V a ïsa \" l a \" , Quasiconformal maps and positive boundary measure, Analysis, 9(1-2):205-216, 1989. · Zbl 0674.30018
[52] S. K. Vodop’yanov, Topological and geometric properties of mappings with an inte- grable Jacobian in Sobolev classes. I, Sibirsk. Mat. Zh., 41(1):23-48, i, 2000. · Zbl 0983.30009
[53] S. K. Vodop’yanov - V. M. Gol’dshtei\?n - Y. G. Reshetnyak, The geometric properties of functions with generalized first derivatives, Uspekhi Mat. Nauk, 34(1(205)):17-65, 1979.
[54] K. Wildrick - T. Zu\"rcher, Peano cubes with derivatives in a Lorentz space, Illinois J. Math., 53(2):365-378, 2009. · Zbl 1223.46038
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