Nonlinear analysis and quasiconformal mappings from the perspective of PDEs
Banach Center Publications (1999)
- Volume: 48, Issue: 1, page 119-140
- ISSN: 0137-6934
Access Full Article
topAbstract
topHow to cite
topIwaniec, Tadeusz. "Nonlinear analysis and quasiconformal mappings from the perspective of PDEs." Banach Center Publications 48.1 (1999): 119-140. <http://eudml.org/doc/208947>.
@article{Iwaniec1999,
abstract = {Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp $L^p$-estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136},
author = {Iwaniec, Tadeusz},
journal = {Banach Center Publications},
keywords = {nonlinear analysis; quasiregular mappings; quasiconformal mappings; systems of PDEs; generalizations of analytic functions; Cauchy-Riemann equations},
language = {eng},
number = {1},
pages = {119-140},
title = {Nonlinear analysis and quasiconformal mappings from the perspective of PDEs},
url = {http://eudml.org/doc/208947},
volume = {48},
year = {1999},
}
TY - JOUR
AU - Iwaniec, Tadeusz
TI - Nonlinear analysis and quasiconformal mappings from the perspective of PDEs
JO - Banach Center Publications
PY - 1999
VL - 48
IS - 1
SP - 119
EP - 140
AB - Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp $L^p$-estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136
LA - eng
KW - nonlinear analysis; quasiregular mappings; quasiconformal mappings; systems of PDEs; generalizations of analytic functions; Cauchy-Riemann equations
UR - http://eudml.org/doc/208947
ER -
References
top- [A] S. S. Antman, Fundamental Mathematical Problems in the Theory of Nonlinear Elasticity, North-Holland, (1976), 33-54.
- [AD] J.J. Alibert and B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions, Arch. Rat. Mech. Anal. 117 (1992), 155-166. Zbl0761.26009
- [As] K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37-60. Zbl0815.30015
- [B] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337-403. Zbl0368.73040
- [Ba] Z. M. Balogh, Jacobians in Sobolev spaces, preprint.
- [Bo] B. Bojarski, Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coeffieients, Mat. Sbornik 43 (1957), 451-503.
- [Be] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153-206. Zbl0756.46017
- [BG] L. Boccardo and T. Gallouet, Non linear elliptic and parabolic equations involving measure data, Jour. Func. Anal. 87 (1989), 149-169. Zbl0707.35060
- [BBH] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhäuser, 19.
- [BK] D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal. 8 (1998), 273-282. Zbl0902.26004
- [BL] R. Bañuelos and A. Lindeman, A martingale study of the Beurling-Ahlfors transform in , Journal of Funct. Anal. 145 (1997), 224-265. Zbl0876.60026
- [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in , Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257-324. Zbl0548.30016
- [BM-S] A. Baernstein and S. J. Montgomery-Smith, Some conjectures about integral means of ∂f and , preprint. Zbl0966.30001
- [BN1] H. Brezis and L. Nirenberg, Degree theory and BMO. Part I: Compact manifolds without boundaries, Selecta Math. 1 (1995), 197-263. Zbl0852.58010
- [BN2] H. Brezis and L. Nirenberg, Degree theory and BMO. Part II: Manifolds with boundaries, Selecta Math. 2 (1996), 309-368. Zbl0868.58017
- [BIS] L. Budney, T. Iwaniec and B. Stroffolini, Removability of singularities of A-harmonic functions, Differential and Integral Equations 12 (1999), 261-274. Zbl1064.35505
- [Bu] D. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158, (1988), 75-94.
- [BW] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575-600. Zbl0853.60040
- [CG] R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators I, Revista Matematica Iberoamericana 8 (1992), 45-67. Zbl0785.47025
- [CJMR] M. Cwikel, B. Jawerth, M. Milman and R. Rochberg, Differential estimates and commutators in interpolation theory, London Math. Soc. Lecture Notes 138 (1989), 170-220. Zbl0696.46050
- [CLMS] R. R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), 247-286. Zbl0864.42009
- [Da] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin 1989. Zbl0703.49001
- [D] G. David, Solutions de l’équation de Beltrami avec , Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), 25-70.
- [DM] B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. I.H.P. Analyse Non Linéaire 7 (1990), 1-26. Zbl0707.35041
- [DHM 1] G. Dolzmann, N. Hungerbühler and S. Müller, Non-linear elliptic systems with measure-valued right hand side, Math. Z. 226 (1997), 545-574. Zbl0895.35029
- [DHM 2] G. Dolzmann, N. Hungerbühler and S. Müller, The p-harmonic system with measure-valued right hand side, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 353-364. Zbl0879.35052
- [DS] S. K. Donaldson and D. B. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163 (1989), 181-252.
- [EH] A. Eremenko and D. Hamilton, On the area distortion by quasiconformal mappings, Proc. Amer. Math. Soc. 123 (1995), 2793-2797. Zbl0841.30013
- [EM] L. C. Evans and S. Müller, Hardy spaces and the two-dimensional Euler equations with nonegative vorticity, J. Amer. Math. Soc. 7 (1994), 199-219. Zbl0802.35120
- [EsM] M. J. Esteban and S. Müller, Sobolev maps with integer degree and applications to Skyrme's problem, Proc. Roy. Soc. London 436 A (1992), 197-201. Zbl0757.49010
- [G] L. Greco, A remark on the equality det Df=Det Df, Diff. Int. Eq. 6 (1993), 1089-1100. Zbl0784.49013
- [Ge] F. W. Gehring, The -integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. Zbl0258.30021
- [GI] F. W. Gehring and T. Iwaniec, The limit of mappings with finite distortion, Ann. Acad. Sci. Fenn. 24 (1999), to appear. Zbl0929.30016
- [GIM] L. Greco, T. Iwaniec and G. Moscariello, Limits of the improved integrability of the volume forms, Indiana University Math. J. 44 (1995), 305-339. Zbl0855.42009
- [GIS] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Mathematica 92 (1997), 249-258. Zbl0869.35037
- [GISS] L. Greco, T. Iwaniec, C. Sbordone and B. Stroffolini, Degree formulas for maps with nonintegrable Jacobian, Topological Methods in Nonlinear Analysis 6 (1995), 81-95. Zbl0854.58005
- [GMS] J. M. Giaquinta, G. Modica and J. Souček, Remarks on the degree theory, J. Funct. Anal. 125 (1994), 172-200. Zbl0822.55003
- [HKM] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford 1993. Zbl0780.31001
- [HK] J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatation, Arch. Rat. Mech. Anal. 125 (1993), 81-97. Zbl0792.30016
- [H] A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), 205-222. Zbl0856.30017
- [I 1] T. Iwaniec, p-harmonic tensors and quasiregular mappings, Ann. of Math. 136, (1992) 589-624. Zbl0785.30009
- [I 2] T. Iwaniec, Integrability theory of the Jacobians, Lipschitz Lectures, preprint No. 36, Sonderforschungsbereich 256, Bonn 1995, pp. 1-68.
- [I 3] T. Iwaniec, Current advances in quasiconformal geometry and nonlinear analysis, Proceedings of the XVIth Rolf Nevanlinna Colloquium, Eds.: Laine/Martio, Walter de Gruyter & Co. (1996), pp. 59-80. Zbl0876.30025
- [I 4] T. Iwaniec, The failure of lower semicontinuity for the linear dilatation, Bull. London Mathematical Society 30 (1998), 55-61. Zbl0924.30031
- [I 5] T. Iwaniec, Nonlinear commutators and Jacobians, Lectures in El Escorial (Spain, 1996), special issue of the Journal of Fourier Analysis and Applications dedicated to Miguel de Guzman, Vol. 3 (1997), 775-796.
- [I 6] T. Iwaniec, Nonlinear Cauchy-Riemann operators in , Proc. AMS, to appear.
- [IL 1] T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rat. Mech. Anal. 125 (1993), 25-79. Zbl0793.58002
- [IL 2] T. Iwaniec and A. Lutoborski, Polyconvex functionals for nearly conformal deformations, SIAM J. Math. Anal., Vol. 27, No. 3 (1996), pp. 609-619. Zbl0847.30014
- [IM 1] T. Iwaniec and G. Martin, Quasiconformal mappings in even dimensions, Acta Math. 170 (1993), 29-81. Zbl0785.30008
- [IM 2] T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. reine angew. Math. 473 (1996), 25-57. Zbl0847.42015
- [IMNS] T. Iwaniec, L. Migliaccio, L. Nania and C. Sbordone, Integrability and removability results for quasiregular mappings in high dimensions, Math. Scand. 75 (1994), 263-279. Zbl0824.30009
- [IMS] T. Iwaniec, M. Mitrea and C. Scott, Boundary value estimates for harmonic forms, Proc. Amer. Math. Soc. 124 (1996), pp. 1467-1471. Zbl0854.31002
- [ISS] T. Iwaniec, C. Scott and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica pura ed Applicata, to appear. Zbl0963.58003
- [IŠ] T. Iwaniec and V. Šverak, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), 181-188. Zbl0784.30015
- [IS 1] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal. 119 (1992), 129-143. Zbl0766.46016
- [IS 2] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161.
- [IS 3] T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO-coefficients, Journal d'Analyse Mathématique 74 (1998), 183-212. Zbl0909.35039
- [IV 1] T. Iwaniec and A. Verde, A study of Jacobians in Hardy-Orlicz spaces, Proc. Royal Soc. Edinburgh (1999).
- [IV 2] T. Iwaniec and A. Verde, Note on the operator 𝓛(f)=f log|f|, submitted to the Journal of Functional Analysis.
- [Ka1] N. J. Kalton, Differential methods in interpolation theory, notes of the lectures at the University of Arkansas, April 10-13, 1996.
- [Ka2] N. J. Kalton, Nonlinear commutators in interpolation theory, Memoirs AMS 385, vol. 73, (1988), 1-85.
- [K] T. Kilpeläinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa 19 (1992), 591-613. Zbl0797.35052
- [L] J. Lewis, On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations 18 (1993), 1515-1537. Zbl0796.35061
- [Li] A. Lindeman, Martingales and the n-dimensional Beurling-Ahlfors transform, preprint (1996).
- [LM] H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton Univ. Press, Princeton, 1989.
- [McM] C. T. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8 (1998), 304-314. Zbl0941.37030
- [Mi] M. Milman, Higher order commutator in the real method of interpolation, Journal D'Analyse Math. 66 (1995), 37-56.
- [M] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. Zbl0141.19407
- [MR] M. Milman and R. Rochberg, The role of cancellation in interpolation theory, Contemporary Math. 189 (1995), 403-410. Zbl0902.46048
- [MS] M. Milman and T. Schonbek, A note on second order estimates in interpolation theory and applications, Proc. AMS 110 (1990), 961-969. Zbl0717.46066
- [Mu1] F. Murat, Compacité par compensations, Ann. Sc. Norm. Sup. Pisa 5 (1978), 489-507. Zbl0399.46022
- [Mu2] F. Murat, Soluciones renormalizades de EDP elipticas no linear, Publications du Laboratoire d'Analyse Numerique, Paris (1993).
- [Ma] J. Manfredi, Quasiregular mappings from the multilinear point of view, Ber. University, Jyväskylä Math. Inst. 68 (1995), 55-94. Zbl0843.30020
- [MV] J. Manfredi and E. Villamor, Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc. 32 (1995), 235-240. Zbl0857.30020
- [Mü] S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. 21 (1989), 245-248. Zbl0689.49006
- [R] H. M. Reimann, Harmonische Funktionen und Jacobi-Determinanten von Diffeomorphismen, Comment. Math. Helv. 47 (1972), 397-408.
- [RRT] J. W. Robbin, R. C. Roger and B. Temple, On weak continuity and the Hodge decomposition, Trans. AMS 303 (1987), 609-618. Zbl0634.35005
- [RY] T. Rivere and D. Ye, Resolutions of the prescribed volume form equation, Nonlinear Differential Equations Appl. 3 (1996), 323-369..
- [Ri] S. Rickman, Quasiregular Mappings, Springer-Verlag, Berlin 1993.
- [Re 1] Y. G. Reshetnyak, On extremal properties of mappings with bounded distortion, Sibirsk. Mat. Z. 10 (1969), 1300-1310 (Russian).
- [Re 2] Y.G. Reshetnyak, Space Mappings with Bounded Distortion, Trans. Math. Monographs 73, Amer. Math. Soc., 1989.
- [Ro] R. Rochberg, Higher order estimates in complex interpolation theory, Pacific J. Math. 174 (1996), 247-267. Zbl0866.46047
- [RW] R. Rochberg and G. Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. 118 (1983), 315-347. Zbl0539.46049
- [S] E. M. Stein, Note on the class LlogL, Studia Math. 32 (1969), 305-310. Zbl0182.47803
- [Sc] C. Scott, -theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995), 2075-2096. Zbl0849.58002
- [Se] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Commun. in PDEs 19 (1994), 277-319. Zbl0836.35030
- [Š] V. Šverak, Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh 120A (1992), 185-189. Zbl0777.49015
- [Ta] L. Tartar, Compensated compactness and applications to partial differential equations in Nonlinear Analysis and Mechanics, Heriotwatt Symposium IV, Research Notes in Mathematics, R. J. Knops (ed.), vol. 39, Pitman, London, (1979), 136-212.
- [T 1] P. Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318-346. Zbl0603.30026
- [T 2] P. Tukia, Compactness properties of μ-homeomorphisms, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991) 47-69.
- [V 1] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, 229, Springer-Verlag, 1971. Zbl0221.30031
- [V 2] J. Väisälä, Questions on quasiconformal maps in space, in: Quasiconformal Mappings and Analysis, Springer, New York, 1998, 369-374. Zbl0892.30020
- [VG] S. K. Vodop'yanov and V. M. Goldstein, Quasiconformal mappings and spaces of functions with generalized first derivatives, Sibirsk. Mat. Z. 17 (1976), 515-531 (Russian).
- [Vu] M. Vuorinen (ed.), Quasiconformal Space Mappings, A collection of Surveys 1960-1990 (Lecture Notes in Math. 1508), Springer-Verlag, Berlin 1992.
- [W] W. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344. Zbl0181.11501
- [Ye] D. Ye, Prescribing the Jacobian determinant in Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 275-296. Zbl0834.35047
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.