Abstract
We show that each quasi-light mapping f in the Sobolev space W 1n(Ω, R n) satisfying ¦Df(x)¦n ≦K(x, f)J(x, f) for almost every x and for some KεL r(Ω), r>n-1, is open and discrete. The assumption that f be quasilight can be dropped if, in addition, it is required that fε W 1p(ω, R n) for some p > = n + 1/ (n-2). More generally, we consider mappings in the John Ball classes Axxx p,q (Ω), and give conditions that guarantee their discreteness and openness.
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References
Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1978), 337–403.
Ball, J. M., Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Royal Soc. Edinburgh 88A (1981), 315–328.
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.
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.
Gol'dshtein, V. M. & S. K. Vodop'yanov, Quasiconformal mappings and spaces of functions with generalized derivatives, Sibirsk. Mat. Z. 17 (1976), 515–531.
Heinonen, J., T. Kilpeläinen & O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, 1993.
Hewitt, E. & K. Stromberg, Real and abstract analysis, Springer-Verlag, Berlin-Heidelberg, 1965.
Iwaniec, T. & C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129–143.
Iwaniec, T. & V. Šverák, On mappings with integrable dilatation, to appear in Proc. Amer. Math. Soc.
Malý, J. & O. Martio, Lusin's condition (N) and mappings of the class W 1n, Preprint, the University of Jyväskylä (1992).
Manfredi, J. J., Weakly monotone functions, preprint (1993).
Martio, O. & W. P. Ziemer, Lusin's condition (N) and mappings with nonnegative Jacobians, Michigan Math. J. 39 (1992), 495–508.
Müller, S., Q. Tang & B. S. Yang, On a new class of elastic deformations not allowing for cavitation, Ann. l'Inst. H. Poincaré, Analyse non linéaire (to appear).
Reshetnyak, Yu. G., Space mappings with bounded distortion, Translation of Mathematical Monographs 73, American Mathematical Society, Providence, 1989.
Rickman, S., Quasiregular mappings, Springer-Verlag, to appear.
Šverák, V., Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal. 100 (1988), 105–127.
Titus, C. J. & G. S. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329–340.
Väisälä, J., Minimal mappings in euclidean spaces, Ann. Acad. Sci. Fenn. Ser. A I 366 (1965), 1–22.
Väisälä, J., Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math. 229, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
Ziemer, W. P., Weakly Differentiate Functions, Springer-Verlag, New York, 1989.
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Communicated by J. M. Ball
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Heinonen, J., Koskela, P. Sobolev mappings with integrable dilatations. Arch. Rational Mech. Anal. 125, 81–97 (1993). https://doi.org/10.1007/BF00411478
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DOI: https://doi.org/10.1007/BF00411478