Abstract
Let \(\Omega \subset {\mathbb {R}}^3\) be a domain and let \(f\in BV_{{\text {loc}}}(\Omega ,{\mathbb {R}}^3)\) be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation \(f^{-1}\in BV_{{\text {loc}}}\). The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York 2000
Ball, J.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. Sect. A88(3–4), 315–328, 1981
Cesari, L.: Surface Area Annals of Mathematics Studies, No. 35, p. 595. Princeton University Press, Princeton, NJ 1956
Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97(3), 171–188, 1987
Conti, S., De Lellis, C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)2, 521–549, 2003
Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space \(W^{1, n-1}\). J. Reine Angew. Math. 644, 221–235, 2010
De Lellis, C.: Some fine properties of currents and applications to distributional Jacobians. Proc. R. Soc. Edinb. 132, 815–842, 2002
De Lellis, C.: Some remarks on the distributional Jacobian. Nonlinear Anal. 53, 1101–1114, 2003
D’Onofrio, L., Schiattarella, R.: On the total variations for the inverse of a BV-homeomorphism. Adv. Calc. Var. 6, 321–338, 2013
D’Onofrio, L., Hencl, S., Malý, J., Schiattarella, R.: Note on Lusin \((N)\) condition and the distributional determinant. J. Math. Anal. Appl. 439, 171–182, 2016
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153, 2nd edn. Springer, New York 1996
Federer, H.: Hausdorff measure and Lebesgue area. Proc. Natl. Acad. Sci. USA37, 90–94, 1951
Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Clarendon Press, Oxford 1995
Gustin, W.: Boxing inequalities. J. Math. Mech. 9, 229–239, 1960
Hatcher, A.: Algebraic Topology, p. xii+544. Cambridge University Press, Cambridge 2002
Henao, D., Mora-Corral, C.: Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197, 619–655, 2010
Henao, D., Mora-Corral, C.: Fracture surfaces and the regularity of inverses for BV deformations. Arch. Ration. Mech. Anal. 201, 575–629, 2011
Henao, D., Mora-Corral, C.: Lusin’s condition and the distributional determinant for deformations with finite energy. Adv. Calc. Var. 5, 355–409, 2012
Hencl, S., Kauranen, A., Malý, J.: On distributional adjugate and the derivative of the inverse, to appear in Annales Academiae Scientiarum Fennicae Mathematica. Preprint arXiv:1904.04574
Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal180, 75–95, 2006
Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion Lecture Notes in Mathematics 2096, p. 176. Springer, Berlin 2014
Hencl, S.: Sharpness of the assumptions for the regularity of a homeomorphism. Mich. Math. J. 59, 667–678, 2010
Hencl, S., Koskela, P., Onninen, J.: Homeomorphisms of bounded variation. Arch. Ration. Mech. Anal186, 351–360, 2007
Iwaniec, T., Koskela, P., Onninen, J.: Mappings of finite distortion: monotonicity and continuity. Invent. Math. 144, 507–531, 2001
Iwaniec, T., Martin, G.: Geometric Function Theory and Nonlinear Analysis. Oxford Mathematical Monographs. Clarendon Press, Oxford 2001
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 1999
Müller, S.: \(Det=det\) A remark on the distributional determinant. C. R. Acad. Sci. Paris Series I Math. 311(1), 13–17, 1990
Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation Arch. Ration. Mech. Anal. 131(1), 1–66, 1995
Müller, S., Spector, S.J., Tang, Q.: Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal. 27, 959–976, 1996
Acknowledgements
The authors would like to thank Jan Malý for pointing their interest to Theorem 2.4 and for many valuable comments and for finding the gap in the original proof of Proposition 4.2. The authors would also like to thank to Ulrich Menne for his information about the literature on Lebesgue area and the anonymous referee for careful reading of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Müller
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
SH and AK were supported in part by the ERC CZ grant LL1203 of the Czech Ministry of Education. SH was supported in part by the grant GAČR P201/18-07996S. AK acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014- 0445). The last author was supported by a grant of the Finnish Academy of Science and Letters.
Rights and permissions
About this article
Cite this article
Hencl, S., Kauranen, A. & Luisto, R. Weak Regularity of the Inverse Under Minimal Assumptions. Arch Rational Mech Anal 238, 185–213 (2020). https://doi.org/10.1007/s00205-020-01540-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-020-01540-4