Change of variables formula under minimal assumptions
Hajłasz, Piotr
Colloquium Mathematicae, Tome 66 (1993), p. 93-101 / Harvested from The Polish Digital Mathematics Library
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In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need the a.e. existence of approximate partial derivatives. In Section 3 we consider the general form of the change of variables formula for Sobolev mappings. The author wishes to thank Professor Bogdan Bojarski for many stimulating conversations and suggestions.

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:210177 
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     title = {Change of variables formula under minimal assumptions},
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     volume = {66},
     year = {1993},
     pages = {93-101},
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Hajłasz, Piotr. Change of variables formula under minimal assumptions. Colloquium Mathematicae, Tome 66 (1993) pp. 93-101. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv64i1p93bwm/

[000] [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in Rn, Ann. Acad. Sci. Fenn. Ser. AI Math. 8 (1983), 257-324. | Zbl 0548.30016

[001] [F] H. Federer, Geometric Measure Theory, Springer, 1969. | Zbl 0176.00801

[002] [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York 1983. | Zbl 0562.35001

[003] [GR] V. M. Goldshteĭn and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings, Nauka, Moscow 1983 (in Russian).

[004] [H] P. Hajłasz, Co-area formula, Sobolev mappings and related topics, in preparation. | Zbl 0988.28002

[005] [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. | Zbl 0283.26003

[006] [IM] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Mittag-Leffler Report #19, 1989/90 (to appear in Acta Math.). | Zbl 0785.30008

[007] [K] M. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108. | Zbl 60.0532.03

[008] [M] O. Martio, Lusin's condition (N) and mappings with non-negative Jacobians, preprint. | Zbl 0807.46032

[009] [P1] S. P. Ponomarev, An example of an ACTLp homeomorphism which is not absolutely continuous in Banach sense, Dokl. Akad. Nauk SSSR 201 (1971), 1053-1054 (in Russian).

[010] [P2] S. P. Ponomarev, On the property N for homeomorphisms of class Wp1, Sibirsk. Mat. Zh. 28 (2) (1987), 140-148 (in Russian).

[011] [RR] T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, 1955. | Zbl 0067.03506

[012] [R1] Yu. G. Reshetnyak, Some geometrical properties of functions and mappings with generalized derivatives, Sibirsk. Mat. Zh. 7 (4) (1966), 886-919 (in Russian).

[013] [R2] Yu. G. Reshetnyak, On the condition N for mappings of class Wn,loc1, ibid. 28 (5) (1987), 149-153 (in Russian).

[014] [S] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, 1983. | Zbl 0546.49019

[015] [V] J. Väisälä, Quasiconformal maps and positive boundary measure, Analysis 9 (1989), 205-216. | Zbl 0674.30018

[016] [W] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math. 1 (1951), 143-159. | Zbl 0043.05803