Lebesgue measure and mappings of the Sobolev class $W^{1,n}$

O. Martio

ArticleOriginal scientific text

Title

Lebesgue measure and mappings of the Sobolev class $W^{1, n}$

Authors ¹

Affiliations

Department of Mathematics, University of Helsinki, FIN-00014 University of Helsinki, Finland

Abstract

We present a survey of the Lusin condition (N) for

W^{1, n}

-Sobolev mappings

f : G \to ℝ^{n}

defined in a domain G of

ℝ^{n}

. Applications to the boundary behavior of conformal mappings are discussed.

Keywords

ENG

jacobians of the Sobolev mappings, mappings of the class

W^{1, n}

, Lusin condition (N)

[BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $ℝ^{n}$ , Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257-324.
[C] L. Cesari, Sulle transformazioni continue, Ann. Mat. Pura Appl. 21 (1941), 157-188.
[G] F. W. Gehring, Rings and quasiconformal mappings in space, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 98-105.
[H] P. Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math. 114 (1993), 93-101.
[HKM] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monographs, Clarendon Press, Oxford 1993.
[JM] P. Jones and N. G. Makarov, Density properties of harmonic measure, preprint.
[M] J. Malý, Hölder type quasicontinuity, to appear.
[MM] J. Malý and O. Martio, Lusin's condition (N) and mappings of the class $W^{1, n}$ , to appear.
[MZ] O. Martio and W. P. Ziemer, Lusin's condition (N) and mappings with non-negative Jacobians, Michigan Math. J., to appear.
[Mu] S. Müller, Higher integrability of determinants and weak convergence in $L^{1}$ , J. Reine Angew. Math. 412 (1990), 20-34.
[NP] R. Näkki and B. Palka, Boundary angles, cusps and conformal mappings, Complex Variables Theory Appl. 5 (1986), 165-180.
[P] S. P. Ponomarev, Examples of homeomorphisms in the class $A C T L^{p}$ which do not satisfy the absolute continuity condition of Banach, Dokl. Akad. Nauk SSSR 201 (1971), 1053-1054 (in Russian).
[RR] T. Radó and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer 1955.
[Rei] H. M. Reimann, On the absolute continuity of a surface representation, Comment. Math. Helv. 46 (1971), 44-47.
[Res1] Yu. G. Reshetnyak, The condition (N) for $W^{1}_{n, l o c}$ space mappings, Sibirsk. Mat. Zh. 28 (1987), 149-153 (in Russian).
[Res2] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monographs 73, Amer. Math. Soc., 1989.
[S] S. Saks, Theory of the Integral, Warsaw, 1937.
[V] J. Väisälä, Quasiconformal maps and positive boundary measure, Analysis 9 (1989), 205-216.

Title

Lebesgue measure and mappings of the Sobolev class W1,n

Affiliations

Abstract

Keywords

Bibliography

Lebesgue measure and mappings of the Sobolev class $W^{1, n}$