Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations
DOI:
https://doi.org/10.24193/subbmath.2022.2.15Keywords:
Sobolev classes, Lusin’s (N )-property, finitely Lipschitz mappings, ring Q-homeomorphisms, lower Q-homeomorphisms, Lipschitz continuity, H¨older continuity, bounded variation.Abstract
We study the distortion features of homeomorphisms of Sobolev class loc admitting integrability for p-outer dilatation. We show that such map- pings belong to W 1,n−1, are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower Q-homeomorphisms with appropriate measurable functions Q. This allows us to derive various distortion results like Lipschitz, H¨older, logarithmic H¨older conti- nuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms.
Mathematics Subject Classification (2010): 30C65, 26B35, 46E35.
Received 19 January 2022; Accepted 18 March 2022.
References
Afanas’eva, E., Golberg, A., Finitely bi-Lipschitz homeomorphisms between Finsler manifolds, Anal. Math. Phys., 10(2020), no. 4, Paper no. 48, 16 pp.
Afanas’eva, E.S., Ryazanov, V.I., Salimov, R.R., Toward the theory of Sobolev-class mappings with a critical exponent, (Russian), Ukr. Mat. Visn. 15(2018), no. 2, 154–176, 295; Translation in J. Math. Sci. (N.Y.), 239(2019), no. 1, 1-16.
Bongiorno, D., A regularity condition in Sobolev spaces W 1,p(Rn) with 1 ≤ p < n, Illinois J. Math., 46(2002), no. 2, 557-570.
Cristea, M., On Poleckii-type modular inequalities, Complex Var. Elliptic Equ., 66(2021), no. 11, 1818-1838.
Cristea, M., Boundary behaviour of open, light mappings in metric measure spaces, Annales Fennici Mathematici, 46(2021), no. 2, 1179-1201.
Cs¨ornyei, M., Hencl, S., Maly, J., Homeomorphisms in the Sobolev space W 1,n−1, J. Reine Angew. Math., 644(2010), 221-235.
Farroni, F., Giova, R., Moscariello, G., Schiattarella, R., Homeomorphisms of finite inner distortion: composition operators on Zygmund-Sobolev and Lorentz-Sobolev spaces, Math. Scand., 116(2015), no. 1, 34-52.
Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wis- senschaften, Band 153 Springer-Verlag New York Inc., New York 1969.
Golberg, A., Salimov, R., Topological mappings of integrally bounded p-moduli, Annals of the University of Bucharest (Mathematical Series), 3(LXI)(2012), 49-66.
Golberg, A., Salimov, R., Logarithmic Holder continuity of ring homeomorphisms with controlled p-module, Complex Var. Elliptic Equ., 59(2014), no. 1, 91-98.
Hencl, S., Koskela, P., Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics, 2096. Springer, Cham, 2014.
Hencl, S., Moscariello, G., Passarelli di Napoli, A., Sbordone, C., Bi-Sobolev mappings and elliptic equations in the plane, J. Math. Anal. Appl., 355(2009), no. 1, 22-32.
Il’yutko, D.P., Sevost’yanov, E.A., On open discrete mappings with unbounded characteristic on Riemannian manifolds, (Russian), Mat. Sb., 207(2016), no. 4, 65-112; Trans- lation in Sb. Math., 207(2016), no. 3-4, 537-580.
Iwaniec, T., Martin, G., Geometric Function Theory and Nonlinear Analysis, Clarendon Press, Oxford, 2001.
Iwaniec, T., Onninen, J., Deformations of finite conformal energy: existence and removability of singularities, Proc. Lond. Math. Soc., 100(2010), no. 1, 1-23.
Kud’yavin, V.S., Golberg, A.L., Mean coefficients of quasiconformality of a pair of domains, Ukrainian Math. J., 43(1991), no. 12, 1594-1597.
Maly, J., Absolutely continuous functions of several variables, J. Math. Anal. Appl., 231(1999), no. 2, 59-61.
Martio, O., Ryazanov, V., Srebro, U., Yakubov, E., Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York, 2009.
Moscariello, G., Passarelli di Napoli, A., The regularity of the inverses of Sobolev homeomorphisms with finite distortion, J. Geom. Anal., 24(2014), no. 1, 571-594.
Ponomarev, S.P., The N−1-property of mappings, and Luzin’s (N ) condition, (Russian), Mat. Zametki, 58(1995), no. 3, 411-418, 480; Translation in Math. Notes, 58(1995), no. 3-4, 960-965.
Saks, S., Theory of the Integral, Second revised edition, English Translation by L. C. Young, With two additional notes by Stefan Banach Dover Publications, Inc., New York, 1964.
Salimov, R.R., Lower Q-homeomorphisms with respect to the p-modulus, (Russian), Ukr. Mat. Visn., 12(2015), no. 4, 484-510, 576; Translation in J. Math. Sci. (N.Y.), 218(2016), no. 1, 47-68.
Salimov, R.R., On the finite Lipschitz property of Orlicz-Sobolev classes, (Russian), Vladikavkaz. Mat. Zh., 17(2015), no. 1, 64-77.
Tengvall, V., Differentiability in the Sobolev space W 1,n−1, Calc. Var. Partial Differential Equations, 51(2014), no. 1-2, 381-399.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
