Lower estimates for 𝑝-moduli and Sobolev class mappings

Salimov, R.

doi:10.1090/spmj/1370

Lower estimates for $p$-moduli and Sobolev class mappings
HTML articles powered by AMS MathViewer

by R. R. Salimov
Translated by: A. Plotkin

St. Petersburg Math. J. 26 (2015), 965-984

DOI: https://doi.org/10.1090/spmj/1370

Published electronically: September 21, 2015

PDF | Request permission

Abstract:

Finite distortion homeomorphisms on the plane are studied with the help of the moduli techniques. On that basis, a problem that dates back to M. A. Lavrent′ev is solved; this problem concerns estimating the area of the image of a disk under the mappings mentioned above. The asymptotic behavior of such mappings at a point is studied. A condition ensuring the finite Lipschitz property is found.

References

M. A. Lavrent′ev, Variatsionnyĭ metod v kraevykh zadachakh dlya sistem yravneniĭ èllipticheskogo tipa, Izdat. Akad. Nauk SSSR, Moscow, 1962. MR 0146377
Olli Martio, Vladimir Ryazanov, Uri Srebro, and Eduard Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York, 2009. MR 2466579
Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 97720, DOI 10.1007/BF02404474
F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 139735, DOI 10.1090/S0002-9947-1962-0139735-8
V. I. Ryazanov and E. A. Sevost′yanov, Equicontinuous classes of ring $Q$-homeomorphisms, Sibirsk. Mat. Zh. 48 (2007), no. 6, 1361–1376 (Russian, with Russian summary); English transl., Siberian Math. J. 48 (2007), no. 6, 1093–1105. MR 2397516, DOI 10.1007/s11202-007-0111-4
R. R. Salimov, Absolute continuity on lines and the differentiability of a generalization of quasiconformal mappings, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 5, 141–148 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 5, 977–984. MR 2473774, DOI 10.1070/IM2008v072n05ABEH002425
R. R. Salimov, Estimation of the measure of the image of the ball, Sibirsk. Mat. Zh. 53 (2012), no. 4, 920–930 (Russian, with Russian summary); English transl., Sib. Math. J. 53 (2012), no. 4, 739–747. MR 3013536, DOI 10.1134/S0037446612040155
R. R. Salimov, On finitely Lipschitz space mappings, Sib. Èlektron. Mat. Izv. 8 (2011), 284–295. MR 2876547, DOI 10.1002/mma.1670080119
—, On the Lipschitz property of a class of mappings, Mat. Zametki 94 (2013), no. 4, 591–599; English transl., Math. Notes 94 (2013), no. 4, 559–566.
R. R. Salimov and E. A. Sevost′yanov, Theory of ring $Q$-mappings and geometric function theory, Mat. Sb. 201 (2010), no. 6, 131–158 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 5-6, 909–934. MR 2682368, DOI 10.1070/SM2010v201n06ABEH004096
E. A. Sevost′yanov, On the theory of the removal of singularities for mappings with an unbounded characteristic of quasiconformality, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 1, 159–174 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 1, 151–165. MR 2655240, DOI 10.1070/IM2010v074n01ABEH002483
E. A. Sevost′yanov, On space mappings with integral restrictions on the characteristic, Algebra i Analiz 24 (2012), no. 1, 131–156 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 1, 99–115. MR 3013296, DOI 10.1090/S1061-0022-2012-01233-1
E. A. Sevost′yanov, On the branch points of mappings with an unbounded characteristic of quasiconformality, Sibirsk. Mat. Zh. 51 (2010), no. 5, 1129–1146 (Russian, with Russian summary); English transl., Sib. Math. J. 51 (2010), no. 5, 899–912. MR 2768509, DOI 10.1007/s11202-010-0090-8
Anatoly Golberg, Homeomorphisms with integrally restricted moduli, Complex analysis and dynamical systems IV. Part 1, Contemp. Math., vol. 553, Amer. Math. Soc., Providence, RI, 2011, pp. 83–98. MR 2868590, DOI 10.1090/conm/553/10934
V. I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. (N.S.) 130(172) (1986), no. 2, 185–206, 284 (Russian); English transl., Math. USSR-Sb. 58 (1987), no. 1, 185–205. MR 854971, DOI 10.1070/SM1987v058n01ABEH003099
S. K. Vodop′yanov and A. D. Ukhlov, Superposition operators in Sobolev spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (2002), 11–33 (Russian); English transl., Russian Math. (Iz. VUZ) 46 (2002), no. 10, 9–31 (2003). MR 1956487
S. K. Vodop′yanov and A. D. Ukhlov, Superposition operators in Lebesgue spaces and the differentiability of quasi-additive set functions, Vladikavkaz. Mat. Zh. 4 (2002), no. 1, 11–33 (Russian, with Russian summary). MR 2065083
S. K. Vodop′yanov, Description of substitution operators of Sobolev spaces, Dokl. Akad. Nauk 400 (2005), no. 1, 11–15 (Russian). MR 2159591
S. K. Vodopyanov, Composition operators on Sobolev spaces, Complex analysis and dynamical systems II, Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 401–415. MR 2175907, DOI 10.1090/conm/382/07079
D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, Boundary behavior and the Dirichlet problem for the Beltrami equations, Algebra i Analiz 25 (2013), no. 4, 101–124 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 4, 587–603. MR 3184618, DOI 10.1090/S1061-0022-2014-01308-8
Tatyana Lomako, Ruslan Salimov, and Evgeny Sevostyanov, On equicontinuity of solutions to the Beltrami equations, Ann. Univ. Buchar. Math. Ser. 1(LIX) (2010), no. 2, 263–274. MR 2920236
Vladimir Ryazanov, Ruslan Salimov, Uri Srebro, and Eduard Yakubov, On boundary value problems for the Beltrami equations, Complex analysis and dynamical systems V, Contemp. Math., vol. 591, Amer. Math. Soc., Providence, RI, 2013, pp. 211–242. MR 3155690, DOI 10.1090/conm/591/11839
Elena S. Afanas′eva, Vladimir I. Ryazanov, and Ruslan R. Salimov, On mappings in Orlicz-Sobolev classes on Riemannian manifolds, Ukr. Mat. Visn. 8 (2011), no. 3, 319–342, 461 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 181 (2012), no. 1, 1–17. MR 2906695, DOI 10.1007/s10958-012-0672-z
D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost′yanov, On the theory of Orlicz-Sobolev classes, Algebra i Analiz 25 (2013), no. 6, 50–102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 6, 929–963. MR 3234840, DOI 10.1090/s1061-0022-2014-01324-6
Denis Kovtonyuk, Vladimir Ryazanov, Ruslan Salimov, and Evgeny Sevost’yanov, On mappings in the Orlicz-Sobolev classes, Ann. Univ. Buchar. Math. Ser. 3(LXI) (2012), no. 1, 67–78. MR 3034963
O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969), 40. MR 0259114
Joseph Hesse, A $p$-extremal length and $p$-capacity equality, Ark. Mat. 13 (1975), 131–144. MR 379871, DOI 10.1007/BF02386202
V. A. Shlyk, On the equality between $p$-capacity and $p$-modulus, Sibirsk. Mat. Zh. 34 (1993), no. 6, 216–221, v, x (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 6, 1196–1200. MR 1268174, DOI 10.1007/BF00973485
F. W. Gehring, Quasiconformal mappings, Complex analysis and its applications (Lectures, Internat. Sem., Trieste, 1975) Internat. Atomic Energy Agency, Vienna, 1976, pp. 213–268. MR 0480997
V. M. Gol′dshteĭn and Yu. G. Reshetnyak, Vvedenie v teoriyu funktsiĭ s obobshchennymi proizvodnymi i kvazikonformnye otobrazheniya, “Nauka”, Moscow, 1983 (Russian). MR 738784
V. G. Maz′ya, Classes of domains, measures and capacities in the theory of spaces of differentiable functions, Current problems in mathematics. Fundamental directions, Vol. 26 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 159–228, 237 (Russian, with Russian summary). MR 1178112
Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
Denis A. Kovtonyuk and Vladimir I. Ryazanov, On the theory of lower $Q$-homeomorphisms, Ukr. Mat. Visn. 5 (2008), no. 2, 159–184, 288 (Russian, with Russian summary); English transl., Ukr. Math. Bull. 5 (2008), no. 2, 157–181. MR 2559832
William P. Ziemer, Extremal length and $p$-capacity, Michigan Math. J. 16 (1969), 43–51. MR 247077
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
T. Rado and P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. MR 0079620
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
F. W. Gehring and Olli Lehto, On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. Ser. A I 272 (1959), 9. MR 0124487
F. W. Gehring, Lipschitz mappings and the $p$-capacity of rings in $n$-space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 175–193. MR 0310238
Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR 1859913
Tadeusz Iwaniec and Vladimír Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), no. 1, 181–188. MR 1160301, DOI 10.1090/S0002-9939-1993-1160301-5
Denis Kovtonyk and Vladimir Ryazanov, On the theory of mappings with finite area distortion, J. Anal. Math. 104 (2008), 291–306. MR 2403438, DOI 10.1007/s11854-008-0025-5
V. G. Maz′ya, Prostranstva S. L. Soboleva, Leningrad. Univ., Leningrad, 1985 (Russian). MR 807364
D. Menchoff, Sur les différentielles totales des fonctions univalentes, Math. Ann. 105 (1931), no. 1, 75–85 (French). MR 1512705, DOI 10.1007/BF01455809