On BV Homeomorphisms

Luigi D'Onofrio; Jan Malý; Carlo Sbordone; Roberta Schiattarella

doi:10.68381/jca28031

Abstract

We obtain the rectifiability of the graph of a bounded variation homeomorphism f in the plane and relations between gradients of f and its inverse. Further, we show an example of a bounded variation homeomorphism f in the plane which satisfies the (N) and

(N^{-1})

properties and strict positivity of Jacobian of both itself and its inverse, but neither f nor

f^{-1}

is Sobolev.

Author information

Contact details are reproduced from the original publication and may be historical.

Luigi D'Onofrio

Dip. di Scienze e Tecnologie, Università degli Studi di Napoli "Parthenope", Napoli, Italy

donofrio@uniparthenope.it

Jan Malý

Dept. of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

maly@karlin.mff.cuni.cz

Carlo Sbordone

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, Italy

sbordone@unina.it

Roberta Schiattarella

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, Italy

roberta.schiattarella@unina.it

Keywords

Bounded variation homeomorphisms
rectifiability of the graph
distributional Jacobian

Mathematics Subject Classification

26B30, 49Q15, 58A25, 46E35

References

G. Alberti: Rank one property for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 239–274.https://doi.org/10.1017/S030821050002566X Google Scholar
L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar
J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Analysis 63 (1977) 337–403.https://doi.org/10.1007/BF00279992 Google Scholar
M. Csörnyei, S. Hencl, J. Malý: Homeomorphisms in the Sobolev space W 1,n−1, J. Reine Angew. Math. 644 (2010) 221–235.https://doi.org/10.1515/crelle.2010.057 Google Scholar
L. D’Onofrio, S. Hencl, J. Malý, R. Schiattarella: Note on Lusin (N) condition and the distributional determinant, J. Math. Analysis Appl. 439 (2016) 171–182.https://doi.org/10.1016/j.jmaa.2016.01.002 Google Scholar
L. D’Onofrio, R. Schiattarella: On the total variations for the inverse of a BV-homeomorphism, Adv. Calc. Var. 6 (2013) 321–338.https://doi.org/10.1515/acv-2012-0103 Google Scholar
H. Federer: Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften 153, 2nd ed., Springer, New York (1996).Google Scholar
I. Fonseca, W. Gangbo: Degree Theory in Analysis and Applications, Clarendon Press, Oxford (1995).https://doi.org/10.1093/oso/9780198511960.001.0001 Google Scholar
M. Giaquinta, G. Modica, J. Souček: Cartesian Currents in the Calculus of Variations. I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete 37, Springer, Berlin (1998).https://doi.org/10.1007/978-3-662-06218-0 Google Scholar
S. Hencl, A. Kauranen, R. Luisto: Weak regularity of the inverse under minimal assumptions, Arch. Rational Mech. Analysis 238 (2020) 185–213.https://doi.org/10.1007/s00205-020-01540-4 Google Scholar
S. Hencl, A. Kauranen, J. Malý: On distributional adjugate and derivative of the inverse, Ann. Acad. Sci. Fenn., arXiv: 1904.04574 (2019).Google Scholar
S. Hencl, P. Koskela: Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Rational Mech. Analysis 180 (2006) 75–95.https://doi.org/10.1007/s00205-005-0394-1 Google Scholar
S. Hencl, P. Koskela: Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics 2096, Springer, Berlin (2014).https://doi.org/10.1007/978-3-319-03173-6 Google Scholar
S. Hencl, P. Koskela, J. Malý: Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. A 136/6 (2006) 1267–1285.https://doi.org/10.1017/S0308210500004972 Google Scholar
S. Hencl, P. Koskela, J. Onninen: Homeomorphisms of bounded variation, Arch. Rational Mech. Analysis 186 (2007) 351–360.https://doi.org/10.1007/s00205-007-0056-6 Google Scholar
S. Hencl, G. Moscariello, A. Passarelli di Napoli, C. Sbordone: Bi-Sobolev mappings and elliptic equations in the plane, J. Math. Analysis Appl. 355/1 (2009) 22–32.https://doi.org/10.1016/j.jmaa.2009.01.026 Google Scholar
J. Kauhanen, P. Koskela, J. Malý: On functions with derivatives in a Lorentz space, Manuscripta Mathematica 100/1 (1999) 87–101.https://doi.org/10.1007/s002290050197 Google Scholar
F. Maggi: Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, Cambridge (2012).https://doi.org/10.1017/CBO9781139108133 Google Scholar
J. Malý, O. Martio: Lusin’s condition (N) and mappings of the class W 1,n, J. Reine Angew. Math. 458 (1995) 19–36.https://doi.org/10.1515/crll.1995.458.19 Google Scholar
J. Malý, D. Swanson, W. P. Ziemer: The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355/2 (2003) 477–492.https://doi.org/10.1090/S0002-9947-02-03091-X Google Scholar
M. Marcus, V. J. Mizel: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79/4 (1973) 790–795.https://doi.org/10.1090/S0002-9904-1973-13319-1 Google Scholar
J. Onninen: Regularity of the inverse of spatial mappings with finite distortion, Calc. Var. Partial Diff. Equations 26/3 (2006) 331–341.https://doi.org/10.1007/s00526-006-0009-1 Google Scholar
K. Quittnerová: Functions of Bounded Variation of Several Variables (in Slovak), Master Thesis, Faculty of Mathematics and Physics, Charles University, Prague (2007).Google Scholar
T. Rado, P. V. Reichelderfer: Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, Cambridge (2012).Google Scholar
Yu. G. Reshetnyak: Some geometrical properties of functions and mappings with generalized derivatives (Russian), Sibirsk. Math. Zh. 7 (1966) 886–919.https://doi.org/10.1007/BF00973267 Google Scholar
L. Simon: Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Volume 3, Australian National University, Sydney (1983).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.

[1] G. Alberti: Rank one property for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 239–274.https://doi.org/10.1017/S030821050002566X Google Scholar

[2] L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar

[3] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Analysis 63 (1977) 337–403.https://doi.org/10.1007/BF00279992 Google Scholar

[4] M. Csörnyei, S. Hencl, J. Malý: Homeomorphisms in the Sobolev space W 1,n−1, J. Reine Angew. Math. 644 (2010) 221–235.https://doi.org/10.1515/crelle.2010.057 Google Scholar

[5] L. D’Onofrio, S. Hencl, J. Malý, R. Schiattarella: Note on Lusin (N) condition and the distributional determinant, J. Math. Analysis Appl. 439 (2016) 171–182.https://doi.org/10.1016/j.jmaa.2016.01.002 Google Scholar

[6] L. D’Onofrio, R. Schiattarella: On the total variations for the inverse of a BV-homeomorphism, Adv. Calc. Var. 6 (2013) 321–338.https://doi.org/10.1515/acv-2012-0103 Google Scholar

[7] H. Federer: Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften 153, 2nd ed., Springer, New York (1996).Google Scholar

[8] I. Fonseca, W. Gangbo: Degree Theory in Analysis and Applications, Clarendon Press, Oxford (1995).https://doi.org/10.1093/oso/9780198511960.001.0001 Google Scholar

[9] M. Giaquinta, G. Modica, J. Souček: Cartesian Currents in the Calculus of Variations. I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete 37, Springer, Berlin (1998).https://doi.org/10.1007/978-3-662-06218-0 Google Scholar

[10] S. Hencl, A. Kauranen, R. Luisto: Weak regularity of the inverse under minimal assumptions, Arch. Rational Mech. Analysis 238 (2020) 185–213.https://doi.org/10.1007/s00205-020-01540-4 Google Scholar

[11] S. Hencl, A. Kauranen, J. Malý: On distributional adjugate and derivative of the inverse, Ann. Acad. Sci. Fenn., arXiv: 1904.04574 (2019).Google Scholar

[12] S. Hencl, P. Koskela: Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Rational Mech. Analysis 180 (2006) 75–95.https://doi.org/10.1007/s00205-005-0394-1 Google Scholar

[13] S. Hencl, P. Koskela: Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics 2096, Springer, Berlin (2014).https://doi.org/10.1007/978-3-319-03173-6 Google Scholar

[14] S. Hencl, P. Koskela, J. Malý: Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. A 136/6 (2006) 1267–1285.https://doi.org/10.1017/S0308210500004972 Google Scholar

[15] S. Hencl, P. Koskela, J. Onninen: Homeomorphisms of bounded variation, Arch. Rational Mech. Analysis 186 (2007) 351–360.https://doi.org/10.1007/s00205-007-0056-6 Google Scholar

[16] S. Hencl, G. Moscariello, A. Passarelli di Napoli, C. Sbordone: Bi-Sobolev mappings and elliptic equations in the plane, J. Math. Analysis Appl. 355/1 (2009) 22–32.https://doi.org/10.1016/j.jmaa.2009.01.026 Google Scholar

[17] J. Kauhanen, P. Koskela, J. Malý: On functions with derivatives in a Lorentz space, Manuscripta Mathematica 100/1 (1999) 87–101.https://doi.org/10.1007/s002290050197 Google Scholar

[18] F. Maggi: Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, Cambridge (2012).https://doi.org/10.1017/CBO9781139108133 Google Scholar

[19] J. Malý, O. Martio: Lusin’s condition (N) and mappings of the class W 1,n, J. Reine Angew. Math. 458 (1995) 19–36.https://doi.org/10.1515/crll.1995.458.19 Google Scholar

[20] J. Malý, D. Swanson, W. P. Ziemer: The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355/2 (2003) 477–492.https://doi.org/10.1090/S0002-9947-02-03091-X Google Scholar

[21] M. Marcus, V. J. Mizel: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79/4 (1973) 790–795.https://doi.org/10.1090/S0002-9904-1973-13319-1 Google Scholar

[22] J. Onninen: Regularity of the inverse of spatial mappings with finite distortion, Calc. Var. Partial Diff. Equations 26/3 (2006) 331–341.https://doi.org/10.1007/s00526-006-0009-1 Google Scholar

[23] K. Quittnerová: Functions of Bounded Variation of Several Variables (in Slovak), Master Thesis, Faculty of Mathematics and Physics, Charles University, Prague (2007).Google Scholar

[24] T. Rado, P. V. Reichelderfer: Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, Cambridge (2012).Google Scholar

[25] Yu. G. Reshetnyak: Some geometrical properties of functions and mappings with generalized derivatives (Russian), Sibirsk. Math. Zh. 7 (1966) 886–919.https://doi.org/10.1007/BF00973267 Google Scholar

[26] L. Simon: Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Volume 3, Australian National University, Sydney (1983).Google Scholar