Regularity of mappings inverse to Sobolev mappings

S. K. Vodopyanov

Sbornik. Mathematics, Tome 203 (2012) no. 10, pp. 1383-1410

Résumé

For homeomorphisms $\varphi\colon\Omega\to \Omega'$ on Euclidean domains in $\mathbb R^n$, $n\geqslant2$, necessary and sufficient conditions ensuring that the inverse mapping belongs to a Sobolev class are investigated. The result obtained is used to describe a new two-index scale of homeomorphisms in some Sobolev class such that their inverses also form a two-index scale of mappings, in another Sobolev class. This scale involves quasiconformal mappings and also homeomorphisms in the Sobolev class $W^1_{n-1}$ such that $\operatorname{rank}D\varphi(x)\leqslant n-2$ almost everywhere on the zero set of the Jacobian $\det D\varphi(x)$. Bibliography: 65 titles.

Détail

Keywords: Sobolev class of mappings, approximate differentiability, distortion and codistortion of mappings, generalized quasiconformal mapping, composition operator.

Bibliographie

[1] G. D. Mostow, “Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms”, Inst. Hautes Études Sci. Publ. Math., 34:1 (1968), 53–104 | DOI | MR | Zbl

[2] J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971 | DOI | MR | Zbl

[3] S. L. Sobolev, “O nekotorykh gruppakh preobrazovanii $n$-mernogo prostranstva”, Dokl. AN SSSR, 32:6 (1941), 380–382 | MR | Zbl

[4] V. G. Mazya, Klassy mnozhestv i teoremy vlozheniya funktsionalnykh klassov. Nekotorye problemy teorii ellipticheskikh operatorov, Dis. ... kand. fiz.-matem. nauk, LGU, L., 1961

[5] F. W. Gehring, “Lipschitz mappings and the $p$-capacity of rings in $n$-space”, Advances in the theory of Riemann surfaces (Stony Brook, NY, 1969), Princeton Univ. Press, Princeton, NJ, 1971, 175–193 | MR | Zbl

[6] S. K. Vodop'yanov, V. M. Gol'dshtein, “Lattice isomorphisms of the spaces $W^1_n$ and quasiconformal mappings”, Siberian Math. J., 16:2 (1975), 174–189 | DOI | MR | Zbl | Zbl

[7] S. K. Vodop'yanov, V. M. Gol'dshtein, “Functional characteristics of quasiisometric mappings”, Siberian Math. J., 17:4 (1976), 580–584 | DOI | MR | Zbl | Zbl

[8] V. M. Gol'dshtein, A. S. Romanov, “Transformations that preserve Sobolev spaces”, Siberian Math. J., 25:3 (1984), 382–388 | DOI | MR | Zbl

[9] S. K. Vodop'yanov, “Composition operators on Sobolev spaces”, Complex analysis and dynamical systems II (Nahariya, Israel, 2003), Contemp. Math., 382, Amer. Math. Soc., Providence, RI, 2005, 401–415 | MR | Zbl

[10] S. K. Vodopyanov, “$L_p$-teoriya potentsiala i kvazikonformnye otobrazheniya na odnorodnykh gruppakh”, Sovremennye problemy geometrii i analiza, Nauka, Novosibirsk, 1989, 45–89 | MR | Zbl

[11] N. Dunford, J. T. Schwartz, Linear operators, Pure Appl. Math., 7, Intersci. Publ., New York–London, 1958 | MR | MR | Zbl

[12] M. Nakai, “Algebraic criterion on quasiconformal equivalence of Riemann surfaces”, Nagoya Math. J, 16 (1960), 157–184 | MR | Zbl

[13] L. G. Lewis, “Quasiconformal mappings and Royden algebras in space”, Trans. Amer. Math. Soc., 158:2 (1971), 481–492 | DOI | MR | Zbl

[14] S. K. Vodopyanov, Formula Teilora i funktsionalnye prostranstva, NGU, Novosibirsk, 1988 | MR | Zbl

[15] S. K. Vodop'yanov, “Mappings of homogeneous groups and imbeddings of functional spaces”, Siberian Math. J., 30:5 (1989), 685–698 | DOI | MR | Zbl

[16] S. K. Vodopyanov, Geometricheskie aspekty prostranstv obobschenno-differentsiruemykh funktsii, Dis. ... dokt. fiz.-matem. nauk, In-t matematiki im. S. L. Soboleva, Novosibirsk, 1992

[17] S. K. Vodopyanov, “Vesovye prostranstva Soboleva i teoriya otobrazhenii”, Tez. dokladov Vsesoyuznoi matematicheskoi shkoly “Teoriya potentsiala” (Katsiveli, 1991), In-t matem. AN USSR, Kiev, 1991, 7

[18] S. K. Vodopyanov, Funktsionalno-teoreticheskii podkhod k nekotorym zadacham teorii prostranstvennykh kvazikonformnykh otobrazhenii, Dis. ... kand. fiz.-matem. nauk, Sib. otdelenie AN SSSR, In-t matematiki, Novosibirsk, 1975

[19] S. K. Vodop'yanov, V. M. Gol'dshtein, “Quasiconformal mappings and spaces of functions with generalized first derivatives”, Siberian Math. J., 17:3 (1976), 399–411 | DOI | MR | Zbl

[20] T. Iwaniec, V. Šverák, “On mappings with integrable dilatation”, Proc. Amer. Math. Soc., 118:1 (1993), 181–188 | DOI | MR | Zbl

[21] A. D. Ukhlov, “On mappings generating the embeddings of Sobolev spaces”, Siberian Math. J., 34:1 (1993), 165–171 | DOI | MR | Zbl

[22] S. K. Vodop'yanov, A. D. Ukhlov, “Sobolev spaces and $(P,Q)$-quasiconformal mappings of Carnot groups”, Siberian Math. J., 39:4 (1998), 665–682 | DOI | MR | Zbl

[23] V. M. Miklyukov, “Ob iskazhenii emkosti pri kvazikonformnykh v srednem otobrazheniyakh”, Materialy itog. nauchn. konf. po matem. i mekh. sek. mat., Tomskii gos. un-t, Tomsk, 1972, 43–45

[24] V. I. Kruglikov, “Capacity of condensers and spatial mappings quasiconformal in the mean”, Math. USSR-Sb., 58:1 (1987), 185–205 | DOI | MR | Zbl

[25] T. Radó, P. V. Reichelderfer, Continuous transformations in analysis with an introduction to algebraic topology, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1955 | MR | Zbl

[26] O. Martio, S. Rickman, J. Väisälä, “Definitions for quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. A I No., 448 (1969), 1–40 | MR | Zbl

[27] S. K. Vodopyanov, “Operatory podstanovki prostranstv Soboleva”, Sovremennye problemy teorii funktsii i ikh prilozheniya (Saratov, 2002), Saratov, 2002, 42–43

[28] S. K. Vodop'yanov, A. D. Ukhlov, “Superposition operators in Sobolev spaces”, Russian Math. (Iz. VUZ), 46:10 (2002), 9–31 | MR | Zbl

[29] S. K. Vodop'yanov, S. K. Ukhlov, “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I”, Siberian Adv. Math., 14:4 (2004), 78–125 | MR | MR | Zbl | Zbl

[30] A. S. Romanov, “O zamene peremennoi v prostranstvakh potentsialov Besselya i Rissa”, Funktsionalnyi analiz i matematicheskaya fizika, Sib. otdelenie AN SSSR, In-t matematiki, Novosibirsk, 1985, 117–133 | MR | Zbl

[31] J. M. Ball, “Convexity conditions and existence theorems in nonlinear elasticity”, Arch. Rational Mech. Anal., 63:4 (1976), 337–403 | DOI | MR | Zbl

[32] J. M. Ball, “Global invertibility of Sobolev functions and the interpenetration of matter”, Proc. Roy. Soc. Edinburgh Sect. A, 88:3–4 (1981), 315–328 | MR | Zbl

[33] J. J. Manfredi, E. Villamor, “Mappings with integrable dilatation in higher dimensions”, Bull. Amer. Math. Soc. (N.S.), 32:2 (1995), 235–240 | DOI | MR | Zbl

[34] S. K. Vodop'yanov, “Topological and geometrical properties of mappings with summable Jacobian in Sobolev classes. I”, Siberian Math. J., 41:1 (2000), 19–39 | DOI | MR | Zbl

[35] J. Kauhanen, P. Koskela, J. Malý, “Mappings of finite distortion: discreteness and openness”, Arch. Ration. Mech. Anal., 160:2 (2001), 135–151 | DOI | MR | Zbl

[36] J. Kauhanen, P. Koskela, J. Malý, “Mappings of finite distortion: condition $N$”, Michigan Math. J., 49:1 (2001), 169–181 | DOI | MR | Zbl

[37] J. Kauhanen, P. Koskela, J. Malý, “Mappings of finite distortion: sharp Orlicz-conditions”, Rev. Mat. Iberoamericana, 19:3 (2003), 857–872 | DOI | MR | Zbl

[38] P. Koskela, J. Onninen, “Mappings of finite distortion: capacity and modulus inequalities”, J. Reine Angew. Math., 599 (2006), 1–26 | DOI | MR | Zbl

[39] A. D. Ukhlov, S. K. Vodopyanov, “Mappings with bounded $(P,Q)$ -distortion on Carnot groups”, Bull. Sci. Math., 134:6 (2010), 605–634 | DOI | MR

[40] Yu. G. Reshetnyak, Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR | Zbl

[41] M. Troyanov, S. Vodop'yanov, “Liouville type theorems for mappings with bounded (co)-distortion”, Ann. Inst. Fourier (Grenoble), 52:6 (2002), 1753–1784 | DOI | MR | Zbl

[42] S. K. Vodop'yanov, “Spaces of differential forms and maps with controlled distortion”, Izv. Math., 74:4 (2010), 663–689 | DOI | MR | Zbl

[43] P. Koskela, “Regularity of the inverse of a Sobolev homeomorphism”, Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), v. III, Hindustan Book Agency, New Dehli, 2010, 1411–1416 | MR | Zbl

[44] W. Stepanoff, “Sur les conditions de l'existence de la différentielle totale”, Matem. sb., 32:3 (1925), 511–527 | Zbl

[45] S. Saks, Theory of the integral, Dover Publ., New York, 1937 | MR | Zbl

[46] H. Whitney, “On totally differentiable and smooth functions”, Pacific J. Math., 1 (1951), 143–159 | MR | Zbl

[47] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969 | MR | MR | Zbl | Zbl

[48] M. Giaquinta, G. Modica, J. Souček, Cartesian currents in the calculus of variations, v. I, Ergeb. Math. Grenzgeb. (3), 37, Springer-Verlag, Berlin, 1998 | MR | Zbl

[49] P. Hajłasz, “Change of variables formula under minimal assumptions”, Colloq. Math., 64:1 (1993), 93–101 | MR | Zbl

[50] M. Csörnyei, S. Hencl, Y. Malý, “Homeomorphisms in the Sobolev space $W^{1,n-1}$”, J. Reine Angew. Math., 644 (2010), 221–235 | DOI | MR | Zbl

[51] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, M.–L., 1950 ; I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publ. Co., New York, 1955 | MR | Zbl | MR | Zbl

[52] Yu. A. Peshkichev, “Inverse mappings for homeomorphisms of class BL”, Math. Notes, 53:5 (1993), 520–522 | DOI | MR | Zbl

[53] A. D. Ukhlov, “Differential and geometric properties of sobolev mappings”, Math. Notes, 75:2 (2004), 291–294 | DOI | MR | Zbl

[54] Yu. G. Reshetnyak, “Some geometrical properties of functions and mappings with generalized derivatives”, Math. Notes, 7:4 (1966), 704–732 | DOI | Zbl

[55] S. K. Vodop'yanov, “Differentiability of maps of Carnot groups of Sobolev classes”, Sb. Math., 194:6 (2003), 857–877 | DOI | MR | Zbl

[56] S. Hencl, P. Koskela, “Regularity of the inverse of a planar Sobolev homeomorphism”, Arch. Ration. Mech. Anal., 180:1 (2006), 75–95 | DOI | MR | Zbl

[57] S. P. Ponomarev, “Property $N$ of homeomorphisms of the class $W^1_p$”, Siberian Math. J., 28:2 (1987), 291–298 | DOI | MR | Zbl | Zbl

[58] S. Hencl, P. Koskela, Y. Malý, “Regularity of the inverse of a Sobolev homeomorphism in space”, Proc. Roy. Soc. Edinburgh Sect. A, 132:6 (2006), 1267–1285 | DOI | MR | Zbl

[59] V. Gold'shtein, A. Ukhlov, “Sobolev homeomorphisms and composition operators”, Around the research of Vladimir Maz'ya, v. I, Int. Math. Ser. (N. Y.), 11, Springer-Verlag, New York, 2010, 207–220 | DOI | MR | Zbl

[60] V. Gold'shtein, A. Ukhlov, “About homeomorphisms that induce composition operators on Sobolev spaces”, Complex Var. Elliptic Equ., 55:8–10 (2010), 833–845 | DOI | MR | Zbl

[61] Yu. G. Reshetnyak, “To the theory of Sobolev-type classes of functions with values in a metric space”, Siberian Math. J., 47:1 (2006), 117–134 | DOI | MR | Zbl

[62] V. G. Maz'ya, Sobolev spaces, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1985 | MR | MR | Zbl | Zbl

[63] S. Hencl, P. Koskela, “Mappings of finite distortion: composition operator”, Ann. Acad. Sci. Fenn. Math., 33:1 (2008), 65–80 | MR | Zbl

[64] S. K. Vodop'yanov, “On regularity of mappings inverse to Sobolev mappings”, Dokl. Math., 78:3 (2008), 891–895 | DOI | MR

[65] S. K. Vodop'yanov, “Mappings of finite codistortion and Sobolev classes of functions”, Dokl. Math., 84:2 (2011), 640–644 | DOI | MR | Zbl