×

Sobolev mappings with integrable dilatations. (English) Zbl 0792.30016

A continuous mapping \(f:G \to R^ n\), \(G\) a domain in \(R^ n\), is said to be quasilight if for each \(y\) the components of \(f^{-1}(y)\) are compact. Note that every discrete mapping is quasilight. The authors show that a quasilight mapping \(f \in W^{1,n} (G)\) satisfying \[ | Df(x) |^ n \leq K(x) J(x,f) \text{ a.e. for some } K \in L^ r(G),\;r>n- 1, \tag{*} \] is open and discrete. Yu. G. Reshetnyak [Sib. Math. Zh. 8, 629-658 (1967; Zbl 0162.381)] proved that for \(f \in W^{1,n} (G)\) condition \((*)\) with \(K \in L^ \infty(G)\) guarantees that \(f\) is either constant or discrete and open. This is the fundamental result in the theory of quasiregular (space) mappings. Subsequently T. Iwaniec and V. Šverák: [Proc. Am. Math. Soc. 118, No. 1, 181-188 (1993; Zbl 0784.30015)] showed that for \(n=2\), \(K \in L^ 1(G)\) suffices for this conclusion. The proof of the present authors employs careful analysis of the Hausdorff dimension of \(f^{-1} (y)\) together with some capacity estimates. The authors also show that if \(f \in W^{1,p} (G)\), \(p \geq n+1/(n-2)\), then the quasilight assumption is superfluous.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1978), 337-403. · Zbl 0368.73040
[2] Ball, J. M., Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Royal Soc. Edinburgh 88A (1981), 315-328. · Zbl 0478.46032
[3] Bojarski, B. & 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. · Zbl 0548.30016
[4] Gehring, F. W., The Hausdorff measure of sets which link in Euclidean space, Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers, Academic Press, New York, 1974.
[5] Gol’dshtein, V. M. & S. K. Vodop’yanov, Quasiconformal mappings and spaces of functions with generalized derivatives, Sibirsk. Mat. Z. 17 (1976), 515-531.
[6] Heinonen, J., T. Kilpel?inen & O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, 1993.
[7] Hewitt, E. & K. Stromberg, Real and abstract analysis, Springer-Verlag, Berlin-Heidelberg, 1965. · Zbl 0137.03202
[8] Iwaniec, T. & C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143. · Zbl 0766.46016
[9] Iwaniec, T. & V. ?ver?k, On mappings with integrable dilatation, to appear in Proc. Amer. Math. Soc.
[10] Mal?, J. & O. Martio, Lusin’s condition (N) and mappings of the class W 1n , Preprint, the University of Jyv?skyl? (1992).
[11] Manfredi, J. J., Weakly monotone functions, preprint (1993). · Zbl 0805.35013
[12] Martio, O. & W. P. Ziemer, Lusin’s condition (N) and mappings with nonnegative Jacobians, Michigan Math. J. 39 (1992), 495-508. · Zbl 0807.46032
[13] M?ller, S., Q. Tang & B. S. Yang, On a new class of elastic deformations not allowing for cavitation, Ann. l’Inst. H. Poincar?, Analyse non lin?aire (to appear).
[14] Reshetnyak, Yu. G., Space mappings with bounded distortion, Translation of Mathematical Monographs 73, American Mathematical Society, Providence, 1989.
[15] Rickman, S., Quasiregular mappings, Springer-Verlag, to appear.
[16] ?ver?k, V., Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal. 100 (1988), 105-127. · Zbl 0659.73038
[17] Titus, C. J. & G. S. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329-340. · Zbl 0113.38001
[18] V?is?l?, J., Minimal mappings in euclidean spaces, Ann. Acad. Sci. Fenn. Ser. A I 366 (1965), 1-22. · Zbl 0144.22103
[19] V?is?l?, J., Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math. 229, Springer-Verlag, Berlin-Heidelberg-New York, 1971. · Zbl 0221.30031
[20] Ziemer, W. P., Weakly Differentiate Functions, Springer-Verlag, New York, 1989. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.