×

Limits of Sobolev homeomorphisms. (English) Zbl 1364.30041

Summary: Let \(\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^2\) be topologically equivalent bounded Lipschitz domains. We prove that weak and strong limits of homeomorphisms \(h: \mathbb{X} \overset{\text{onto}} \longrightarrow \mathbb{Y}\) in the Sobolev space \(\mathcal{W}^{1,p} (\mathbb{X}, \mathbb{R}^2)\), \(p \geq 2\), are the same. As an application, we establish the existence of 2D-traction free minimal deformations for fairly general energy integrals.

MSC:

30E10 Approximation in the complex plane
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58E20 Harmonic maps, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alessandrini, G.: Critical points of solutions of elliptic equations in two variables. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14, 229-256 (1987)Zbl 0649.35026 MR 0939628 · Zbl 0649.35026
[2] Alessandrini, G., Nesi, V.: Univalent σ -harmonic mappings. Arch. Ration. Mech. Anal. 158, 155-171 (2001)Zbl 0977.31006 MR 1838656 · Zbl 0977.31006
[3] Alessandrini, G., Sigalotti, M.: Geometric properties of solutions to the anisotropic pLaplace equation in dimension two. Ann. Acad. Sci. Fenn. Math. 26, 249-266 (2001) Zbl 1002.35044 MR 1816571 · Zbl 1002.35044
[4] Antman, S. S.: Nonlinear Problems of Elasticity. Appl. Math. Sci. 107, Springer, New York (1995)Zbl 0820.73002 MR 1323857 · Zbl 0820.73002
[5] Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Univ. Press, Princeton, NJ (2009)Zbl 1182.30001 MR 2472875 · Zbl 1182.30001
[6] Astala, K., Iwaniec, T., Martin, G.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195, 899-921 (2010)Zbl 1219.30011 MR 2591976 · Zbl 1219.30011
[7] Astala, K., Iwaniec, T., Saksman, E.: Beltrami operators in the plane. Duke Math. J. 107, 27-56 (2001)Zbl 1009.30015 MR 1815249 · Zbl 1009.30015
[8] Ball, J. M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337-403 (1976/77)Zbl 0368.73040 MR 0475169 · Zbl 0368.73040
[9] Ball, J. M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, Res. Notes Math. 17, Pitman, London, 187-241 (1977)Zbl 03588136 MR 0478899 · Zbl 0377.73043
[10] Ball, J. M.: Existence of solutions in finite elasticity. In: Finite Elasticity (Betlehem, PA, 1980), Nijhoff, 1-12 (1982)Zbl 0518.73031 · Zbl 0518.73031
[11] Ball, J. M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. London A 306, 557-611 (1982)Zbl 0513.73020 MR 0703623 · Zbl 0513.73020
[12] Ball, J. M.: Minimizers and the Euler-Lagrange equations. In: Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), Lecture Notes in Phys. 195, Springer, Berlin, 1-4 (1984)Zbl 0547.73013 MR 0755716
[13] Ball, J. M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, Springer, New York, 3-59 (2002)Zbl 1054.74008 MR 1919825 · Zbl 1054.74008
[14] Bauman, P., Marini, A., Nesi, V.: Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50, 747-757 (2001) Zbl 1330.35121 MR 1871388 Limits of Sobolev homeomorphisms503 · Zbl 1330.35121
[15] Bauman, P., Owen, N. C., Phillips, D.: Maximum principles and a priori estimates for an incompressible material in nonlinear elasticity. Comm. Partial Differential Equations 17, 1185- 1212 (1992)Zbl 0777.35014 MR 1179283 · Zbl 0777.35014
[16] Bojarski, B.: Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients. Mat. Sb. (N.S.) 43 (85), 451-503 (1957) Zbl 1173.35403 MR 0106324
[17] Bojarski, B., Iwaniec, T.: p-harmonic equation and quasiregular mappings. In: Partial Differential Equations (Warszawa, 1984), Banach Center Publ. 19, PWN, Warszawa, 25-38 (1987) Zbl 0659.35035 MR 1055157 · Zbl 0659.35035
[18] Ciarlet, P. G.: Mathematical Elasticity Vol. I. Three-Dimensional Elasticity. North-Holland Stud. Math. Appl. 20, North-Holland, Amsterdam (1988)Zbl 0648.73014 MR 0936420 · Zbl 0648.73014
[19] Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72, 247-286 (1993)Zbl 0864.42009 MR 1225511 · Zbl 0864.42009
[20] Conti, S., De Lellis, C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2, 521-549 (2003) Zbl 1114.74004 MR 2020859 · Zbl 1114.74004
[21] Cristina, J., Iwaniec, T., Kovalev, L. V., Onninen, J.: The Hopf-Laplace equation: harmonicity and regularity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 13, 1145-1187 (2014) Zbl 1327.30022 MR 3362123 · Zbl 1327.30022
[22] D’Onofrio, L., Iwaniec, T.: Notes on p-harmonic analysis. In: The p-harmonic Equation and Recent Advances in Analysis, Contemp. Math. 370, Amer. Math. Soc., Providence, RI, 25-49 (2005)Zbl 1134.35332 MR 2126700 · Zbl 1134.35332
[23] Duren, P.: Harmonic Mappings in the Plane. Cambridge Tracts in Math. 156, Cambridge Univ. Press, Cambridge (2004)Zbl 1055.31001 MR 2048384 · Zbl 1055.31001
[24] Evans, L. C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95, 227-252 (1986)Zbl 0627.49006 MR 0853966 · Zbl 0627.49006
[25] Hajłasz, P.: Pointwise Hardy inequalities. Proc. Amer. Math. Soc. 127, 417-423 (1999) Zbl 0911.31005 MR 1458875 · Zbl 0911.31005
[26] Hajłasz, P.: Sobolev mappings: Lipschitz density is not a bi-Lipschitz invariant of the target. Geom. Funct. Anal. 17, 435-467 (2007)Zbl 1132.46023 MR 2322491 · Zbl 1132.46023
[27] Heinonen, J., Kilpel¨ainen, T., Mal´y, J.: Connectedness in fine topologies. Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 107-123 (1990)Zbl 0715.31005 MR 1050785 · Zbl 0715.31005
[28] Heinonen, J., Kilpel¨ainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, New York (1993)Zbl 0780.31001 MR 1207810 · Zbl 0780.31001
[29] Iwaniec, T., Koh, N.-T., Kovalev, L. V., Onninen, J.: Existence of energy-minimal diffeomorphisms between doubly connected domains. Invent. Math. 186, 667-707 (2011) Zbl 1255.30031 MR 2854087 · Zbl 1255.30031
[30] Iwaniec, T., Kovalev, L. V., Onninen, J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201, 1047-1067 (2011)Zbl 1260.46023 MR 2824471 · Zbl 1260.46023
[31] Iwaniec, T., Kovalev, L. V., Onninen, J.: Approximation up to the boundary of homeomorphisms of finite Dirichlet energy. Bull. London Math. Soc. 44, 871-881 (2012) Zbl 1264.30025 MR 2975148 · Zbl 1264.30025
[32] Iwaniec, T., Kovalev, L. V., Onninen, J.: Lipschitz regularity for inner-variational equations. Duke Math. J. 162, 643-672 (2013)Zbl 1319.49055 MR 3039677 · Zbl 1319.49055
[33] Iwaniec, T., Manfredi, J. J.: Regularity of p-harmonic functions on the plane. Rev. Mat. Iberoamer. 5, 1-19 (1989)Zbl 0805.31003 MR 1057335 · Zbl 0805.31003
[34] Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford Math. Monogr., Oxford Univ. Press (2001)Zbl 1045.30011 MR 1859913 504Tadeusz Iwaniec, Jani Onninen · Zbl 1045.30011
[35] Iwaniec, T., Onninen, J.: H1-estimates of Jacobians by subdeterminants. Math. Ann. 324, 341-358 (2002)Zbl 1055.42011 MR 1933861 · Zbl 1055.42011
[36] Iwaniec, T., Onninen, J.: Hyperelastic deformations of smallest total energy. Arch. Ration. Mech. Anal. 194, 927-986 (2009)Zbl 1193.74013 MR 2563629 · Zbl 1193.74013
[37] Iwaniec, T., Onninen, J.: Neohookean deformations of annuli, existence, uniqueness and radial symmetry. Math. Ann. 348, 35-55 (2010)Zbl 1214.30011 MR 2657433 · Zbl 1214.30011
[38] Iwaniec, T., Onninen, J.: Deformations of finite conformal energy: existence and removability of singularities. Proc. London Math. Soc. (3) 100, 1-23 (2010)Zbl 1196.30023 MR 2578466 · Zbl 1196.30023
[39] Iwaniec, T., Onninen, J.: Deformations of finite conformal energy: boundary behavior and limit theorems. Trans. Amer. Math. Soc. 363, 5605-5648 (2011)Zbl 1246.30040 MR 2817402 · Zbl 1246.30040
[40] Iwaniec, T., Onninen, J.: n-Harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218, no. 1023 (2012)Zbl 1294.30042 MR 2976798 · Zbl 1294.30042
[41] Iwaniec, T., Onninen, J.: Mappings of least Dirichlet energy and their Hopf differentials. Arch. Ration. Mech. Anal. 209, 401-453 (2013)Zbl 1294.30085 MR 3056614 · Zbl 1294.30085
[42] Iwaniec, T., Onninen, J.: Monotone Sobolev mappings of planar domains and surfaces. Arch. Ration. Mech. Anal. 219, 159-181 (2016)Zbl 06545481 MR 3437849 · Zbl 1395.30023
[43] Jerison, D. S., Kenig, C. E.: Hardy spaces, A∞, and singular integrals on chord-arc domains. Math. Scand. 50, 221-247 (1982)Zbl 0509.30025 MR 0672926 · Zbl 0509.30025
[44] Jost, J.: A note on harmonic maps between surfaces. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 2, 397-405 (1985)Zbl 0585.58011 MR 0831039 · Zbl 0585.58011
[45] Jost, J.: Two-Dimensional Geometric Variational Problems. Wiley, Chichester (1991) Zbl 0729.49001 MR 1100926 · Zbl 0729.49001
[46] Jost, J., Schoen, R.: On the existence of harmonic diffeomorphisms. Invent. Math. 66, 353- 359 (1982)Zbl 0488.58009 MR 0656629 · Zbl 0488.58009
[47] Kilpel¨ainen, T., Mal´y, J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19, 591-613 (1992)Zbl 0797.35052 MR 1205885 · Zbl 0797.35052
[48] Kuratowski, K.: On the completeness of the space of monotone mappings and some related problems. Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 16, 283-285 (1968) Zbl 0155.50201 MR 0232337 · Zbl 0155.50201
[49] Kuratowski, K., Lacher, R. C.: A theorem on the space of monotone mappings. Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 17, 797-800 (1969)Zbl 0195.24401 MR 0275386 · Zbl 0195.24401
[50] Lehrb¨ack, J.: Pointwise Hardy inequalities and uniformly fat sets. Proc. Amer. Math. Soc. 136, 2193-2200 (2008)Zbl 1151.46021 MR 2383525 · Zbl 1151.46021
[51] Lehto, O., Virtanen, K. I.: Quasiconformal Mappings in the Plane. Springer, New York (1973) Zbl 0267.30016 MR 0344463 · Zbl 0267.30016
[52] Lewis, J. L.: On critical points of p-harmonic functions in the plane. Electron. J. Differential Equations 1994, no. 3, 4 pp.Zbl 0808.31008 MR 1281471 · Zbl 0808.31008
[53] Manfredi, J. J.: p-harmonic functions in the plane. Proc. Amer. Math. Soc. 103, 473-479 (1988)Zbl 0658.35041 MR 0943069 · Zbl 0658.35041
[54] Marcus, M., Mizel, V. J.: Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33, 217-229 (1979)Zbl 0418.46024 MR 0546508 · Zbl 0418.46024
[55] Marsden, J. E., Hughes, T. J. R.: Mathematical Foundations of Elasticity. Dover Publ., New York (1994)MR 1262126 Limits of Sobolev homeomorphisms505 · Zbl 0545.73031
[56] McAuley, L. F.: Some fundamental theorems and problems related to monotone mappings. In: Proc. First Conf. on Monotone Mappings and Open Mappings (Binghamton, NY, 1970), State Univ. of New York at Binghamton, NY, 1-36 (1971)Zbl 0226.54006 MR 0287518 · Zbl 0226.54006
[57] Miniowitz, R.: Normal families of quasimeromorphic mappings. Proc. Amer. Math. Soc. 84, 35-43 (1982)Zbl 0478.30024 MR 0633273 · Zbl 0478.30024
[58] Morrey, C. B.: The topology of (path) surfaces. Amer. J. Math. 57, 17-50 (1935) Zbl 0011.03701 MR 1507053 · JFM 61.0640.01
[59] Morrey, C. B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2, 25-53 (1952)Zbl 0046.10803 MR 0054865 · Zbl 0046.10803
[60] M¨uller, S.: Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math. 412, 20-34 (1990)Zbl 0713.49004 MR 1078998 · Zbl 0713.49004
[61] Rad´o, T.: On continuous mappings of Peano spaces. Trans. Amer. Math. Soc. 58, 420-454 (1945)Zbl 0061.10802 MR 0014419 · Zbl 0061.10802
[62] Rad´o, T.: Length and Area. Amer. Math. Soc., New York (1948)Zbl 0033.17002 MR 0024511 · Zbl 0033.17002
[63] Reshetnyak, Yu. G.: Space Mappings with Bounded Distortion. Amer. Math. Soc., Providence, RI (1989)Zbl 0667.30018 MR 0994644 · Zbl 0667.30018
[64] ˇSilhav´y, M.: The Mechanics and Thermodynamics of Continuous Media. Texts Monogr. Phys., Springer, Berlin (1997)Zbl 0870.73004 MR 1423807 · Zbl 0870.73004
[65] Sivaloganathan, J., Spector, S. J.: Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 25, 201-213 (2008)Zbl 1137.74011 MR 2383087 · Zbl 1137.74011
[66] Sivaloganathan, J., Spector, S. J.: On irregular weak solutions of the energy-momentum equations. Proc. Roy. Soc. Edinburgh Sect. A 141, 193-204 (2011)Zbl 1211.49006 MR 2773446 · Zbl 1211.49006
[67] Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (2004) Zbl 0779.73004 MR 2056350 · Zbl 1068.74002
[68] Tukia, P.: The planar Sch¨onflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 49-72 (1980)Zbl 0411.57015 MR 0595177 · Zbl 0411.57015
[69] V¨ais¨al¨a, J.: Homeomorphisms of bounded length distortion. Ann. Acad. Sci. Fenn. Ser. A I Math. 12, 303-312 (1987)Zbl 0653.30009 MR 0951979 · Zbl 0653.30009
[70] Whyburn, G. T.: Analytic Topology. Amer. Math. Soc., Providence, RI (1963) Zbl 0117.15804 MR 0182943 · Zbl 0117.15804
[71] Youngs, J. W. T.: The topological theory of Fr´echet surfaces. Ann. of Math. (2) 45, 753-785 (1944)Zbl 0061.10903 MR 0012222 · Zbl 0061.10903
[72] Youngs, J. W. T.: Homeomorphic approximations to monotone mappings. Duke Math. J. 15, 87-94 (1948)Zbl 0030.41603 MR 0024623 · Zbl 0030.41603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.