Abstract
Let \({\Omega\subset\mathbb{R}^{n}}\) be a domain. We show that each homeomorphism f in the Sobolev space \({W^{1,1}_{\rm loc}(\Omega,\mathbb{R}^{n})}\) satisfies either J f ≥ 0 a.e or J f ≤ 0 a.e. if n = 2 or n = 3. For n > 3 we prove the same conclusion under the stronger assumption that \({f\in W^{1,s}_{\rm loc}(\Omega,\mathbb{R}^{n})}\) for some s > [n/2] (or in the setting of Lorentz spaces).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Astala K., Iwaniec T., Martin G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ (2009)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, Boca Raton (1992)
Fonseca I., Gangbo W.: Degree Theory in Analysis and Applications. Clarendon Press, Oxford (1995)
Gehring F.W., Lehto O.: On the total differentiability of functions of a complex variable. Ann. Acad. Sci. Fenn. Ser. A I 272, 1–9 (1959)
Hajłasz, P.: private communication (2002)
Hencl S., Koskela P., Onninen J.: Homeomorphisms of bounded variation. Arch. Ration. Mech. Anal. 186, 351–360 (2007)
Iwaniec T., Martin G.: Geometric Function Theory and Nonlinear Analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2001)
Iwaniec T., Martin G.: Squeezing the Sierpiñski sponge. Studia Math. 149, 133–145 (2002)
Kauhanen J., Koskela P., Malý J.: On functions with derivatives in a Lorentz space. Manuscripta Math. 100, 87–101 (1999)
Koskela, P.: Lectures on Quasiconformal Mappings. University of Jyväskylä (to appear)
Malý J.: Examples of weak minimizers with continuous singularities. Expos. Math. 13(5), 446–454 (1995)
Rado T., Reichelderfer P.V.: Continuous Transformations in Analysis. Springer, Berlin (1955)
Rickman, S.: Quasiregular mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 26. Springer-Verlag, Berlin (1993)
Rolfsen D.: Knots and Links. Publish or Perish, Berkeley (1976)
Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Swanson D., Ziemer W.P.: A topological aspect of Sobolev mappings. Calc. Var. Partial Differ. Equ. 14(1), 69–84 (2002)
Ziemer W.K.: A note on the support of a Sobolev function on a k-cell. Proc. Am. Math. Soc. 132(7), 1987–1995 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hencl, S., Malý, J. Jacobians of Sobolev homeomorphisms. Calc. Var. 38, 233–242 (2010). https://doi.org/10.1007/s00526-009-0284-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-009-0284-8