Nonlinear analysis and quasiconformal mappings from the perspective of PDEs

• Autorzy: Tadeusz Iwaniec
• Języki publikacji: EN

Abstrakty

EN

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1999
• Tom: 48
• Numer: 1
• Strony: 119-140

Daty

wydano

1999

Twórcy

autor

Tadeusz Iwaniec

• Department of Mathematics, Syracuse University, Syracuse, New York 13244, U.S.A.

Bibliografia

• [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.
• [As] K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37-60.
• [B] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337-403.
• [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.
• [BG] L. Boccardo and T. Gallouet, Non linear elliptic and parabolic equations involving measure data, Jour. Func. Anal. 87 (1989), 149-169.
• [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.
• [BL] R. Bañuelos and A. Lindeman, A martingale study of the Beurling-Ahlfors transform in $R^n$, Journal of Funct. Anal. 145 (1997), 224-265.
• [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $R^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257-324.
• [BM-S] A. Baernstein and S. J. Montgomery-Smith, Some conjectures about integral means of ∂f and $\overline∂f$, preprint.
• [BN1] H. Brezis and L. Nirenberg, Degree theory and BMO. Part I: Compact manifolds without boundaries, Selecta Math. 1 (1995), 197-263.
• [BN2] H. Brezis and L. Nirenberg, Degree theory and BMO. Part II: Manifolds with boundaries, Selecta Math. 2 (1996), 309-368.
• [BIS] L. Budney, T. Iwaniec and B. Stroffolini, Removability of singularities of A-harmonic functions, Differential and Integral Equations 12 (1999), 261-274.
• [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.
• [CG] R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators I, Revista Matematica Iberoamericana 8 (1992), 45-67.
• [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.
• [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.
• [Da] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin 1989.
• [D] G. David, Solutions de l'équation de Beltrami avec $|μ|_∞=1$, 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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [G] L. Greco, A remark on the equality det Df=Det Df, Diff. Int. Eq. 6 (1993), 1089-1100.
• [Ge] F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277.
• [GI] F. W. Gehring and T. Iwaniec, The limit of mappings with finite distortion, Ann. Acad. Sci. Fenn. 24 (1999), to appear.
• [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.
• [GIS] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Mathematica 92 (1997), 249-258.
• [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.
• [GMS] J. M. Giaquinta, G. Modica and J. Souček, Remarks on the degree theory, J. Funct. Anal. 125 (1994), 172-200.
• [HKM] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford 1993.
• [HK] J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatation, Arch. Rat. Mech. Anal. 125 (1993), 81-97.
• [H] A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), 205-222.
• [I 1] T. Iwaniec, p-harmonic tensors and quasiregular mappings, Ann. of Math. 136, (1992) 589-624.
• [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.
• [I 4] T. Iwaniec, The failure of lower semicontinuity for the linear dilatation, Bull. London Mathematical Society 30 (1998), 55-61.
• [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 $ℝ^n$, Proc. AMS, to appear.
• [IL 1] T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rat. Mech. Anal. 125 (1993), 25-79.
• [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.
• [IM 1] T. Iwaniec and G. Martin, Quasiconformal mappings in even dimensions, Acta Math. 170 (1993), 29-81.
• [IM 2] T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. reine angew. Math. 473 (1996), 25-57.
• [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.
• [IMS] T. Iwaniec, M. Mitrea and C. Scott, Boundary value estimates for harmonic forms, Proc. Amer. Math. Soc. 124 (1996), pp. 1467-1471.
• [ISS] T. Iwaniec, C. Scott and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica pura ed Applicata, to appear.
• [IŠ] T. Iwaniec and V. Šverak, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), 181-188.
• [IS 1] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal. 119 (1992), 129-143.
• [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.
• [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.
• [L] J. Lewis, On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations 18 (1993), 1515-1537.
• [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.
• [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.
• [MR] M. Milman and R. Rochberg, The role of cancellation in interpolation theory, Contemporary Math. 189 (1995), 403-410.
• [MS] M. Milman and T. Schonbek, A note on second order estimates in interpolation theory and applications, Proc. AMS 110 (1990), 961-969.
• [Mu1] F. Murat, Compacité par compensations, Ann. Sc. Norm. Sup. Pisa 5 (1978), 489-507.
• [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.
• [MV] J. Manfredi and E. Villamor, Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc. 32 (1995), 235-240.
• [Mü] S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. 21 (1989), 245-248.
• [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.
• [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.
• [RW] R. Rochberg and G. Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. 118 (1983), 315-347.
• [S] E. M. Stein, Note on the class LlogL, Studia Math. 32 (1969), 305-310.
• [Sc] C. Scott, $L^p$-theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995), 2075-2096.
• [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.
• [Š] V. Šverak, Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh 120A (1992), 185-189.
• [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.
• [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.
• [V 2] J. Väisälä, Questions on quasiconformal maps in space, in: Quasiconformal Mappings and Analysis, Springer, New York, 1998, 369-374.
• [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.
• [Ye] D. Ye, Prescribing the Jacobian determinant in Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 275-296.