Nonlinear Cauchy-Riemann operators in

Author:
Tadeusz Iwaniec

Journal:
Trans. Amer. Math. Soc. **354** (2002), 1961-1995

MSC (2000):
Primary 35J60, 30G62; Secondary 42B25, 26B10

DOI:
https://doi.org/10.1090/S0002-9947-02-02914-8

Published electronically:
January 8, 2002

MathSciNet review:
1881026

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

**[AB]**L. V. Ahlfors and A. Beurling,*Conformal invariants and function theoretic null sets*, Acta Math**83**(1950), 101-129. MR**12:171c****[A1]**K. Astala,*Area distortion of quasiconformal mappings*, Acta Math**173**(1994), 37-60. MR**95m:30028b****[A2]**-,*Planar quasiconformal mappings; deformations and interactions*, Quasiconformal Mappings and Analysis, A Collection of Papers Honoring F. W. Gehring (P. Duren, J. Heinonen, B. Osgood and B. Palka, eds.), Springer-Verlag, New York, 1998, 33-54. MR**99a:30025****[A3]**-,*Analytic aspects of quasiconformality*, Proceedings of ICM (Berlin, 1998), Doc. Math.**1998**, Extra Vol. II, 617-626. MR**99h:30024****[AIS]**K. Astala, T. Iwaniec and E. Saksman,*Beltrami operators in the plane*, Duke Math. J.**107**(2001), 27-56. MR**2001m:30021****[B]**J. M. Ball,*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rat. Mech. Anal.**63**(1977), 337-403. MR**57:14788****[BL]**R. Bañuelos and A. Lindeman,*A martingale study of the Beurling-Ahlfors transform in*, J. Funct. Anal.**145**(1997), 224-265. MR**98a:30007****[BM-H]**R. Bañuelos and P. J. Méndez-Hernández,*Space-time Brownian motion and the Beurling-Ahlfors transform*, preprint (2001).**[BM-S]**A. Baernstein and S. J. Montgomery-Smith,*Some conjectures about integral means of**and*, Complex Analysis and Differential Equations (Proc. Sympos. in Honor of Matts Essén, Uppsala, 1997; C. Kiselman and A. Vretblad, eds.), Acta Univ. Upalensis**64**(1999), 92-109. MR**2001i:30002****[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. MR**96k:60108****[B1]**D. L. Burkholder,*Sharp inequalities for martingales and stochastic integrals*, Astérisque**157-158**(1988), 75-94. MR**90b:60051****[B2]**-,*A proof of Pelczynski's conjecture for the Haar system*, Studia Math.**91**(1) (1988), 79-83. MR**89j:46026****[D]**B. Dacorogna,*Direct Methods in the Calculus of Variations*, Applied Mathematical Sciences, vol. 78, Springer-Verlag, 1989. MR**90e:49001****[DS]**S. K. Donaldson and D. P. Sullivan,*Quasiconformal 4-Manifolds*, Acta Math.**163**(1989), 181-252. MR**91d:57012****[G]**F. W. Gehring,*The**-integrability of the partial derivatives of a quasiconformal mapping*, Acta Math.**130**(1973), 265-277. MR**53:5861****[GI]**L. Greco and T. Iwaniec,*New inequalities for the Jacobian*, Ann. Inst. H. Poincaré, Analyse non linéaire**11(1)**(1994), 17-35. MR**95b:42020****[I1]**T. Iwaniec,*Extremal inequalities in Sobolev spaces and quasiconformal mappings*, Z. Anal. Anwendungen**1**(1982), 1-16. MR**85g:30027****[I2]**-,*Projections onto gradient fields and**-estimates for degenerate elliptic operators*, Studia Math.**75**(1983), 293-312. MR**85i:46037****[I3]**-,*-Harmonic tensors and quasiregular mappings*, Annals of Math.**136**(1993), 589-624. MR**94d:30034****[I4]**-,*Integrability theory of the Jacobians*, Lipschitz Lectures in Bonn, preprint 36, Sonderforschungsbereich 256, Bonn, 1995.**[I5]**-,*On the concept of the weak Jacobian and Hessian*, Report of the University of Jyväskylä No 83, dedicated to Olli Martio (2001), 181-205.**[IL]**T. Iwaniec and A. Lutoborski,*Integral estimates for null Lagrangians*, Arch. Rat. Mech. Anal.**125**(1993), 25-79. MR**95c:58054****[IM1]**T. Iwaniec and G. Martin,*Quasiregular mappings in even dimensions*, Acta Math.**170**(1993), 29-81. MR**94m:30046****[IM2]**-,*Riesz transforms and related singular integrals*, J. reine angew. Math.**473**(1996), 25-57. MR**97k:42033****[IM3]**T. Iwaniec and G. Martin,*Geometric Function Theory and Nonlinear Analysis*, Oxford Mathematical Monographs, 2001.**[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. MR**96e:30052****[IS]**T. Iwaniec and C. Sbordone,*Weak minima of variational integrals*, J. reine angew. Math.**454**(1994), 143-161. MR**95d:49035****[IV]**T. Iwaniec and A. Verde,*On the operator*, J. Fun. Anal.**169**(1999), 391-420. MR**2001i:42034****[IVV]**T. Iwaniec, G. Verchota and A. Vogel,*The failure of rank-one connections*, to appear in Arch. Rat. Mech. Anal.**[M]**J. Manfredi,*Quasiregular mappings from the multilinear point of view*, Ber. University Jyväskylä Math. Inst.**68**(1995), 55-94. MR**96j:30034****[M1]**C. B. Morrey,*Quasiconvexity and the semicontinuity of multiple integrals*, Pacific J. Math.**2**(1952), 25-53. MR**14:992a****[M2]**-,*Multiple Integrals in the Calculus of Variations*, Springer-Verlag, Berlin, 1966. MR**34:2380****[MSY]**S. Müller, V. Sverák and B. Yan,*Sharp stability results for almost conformal maps in even dimensions*, Journ. Geom. Anal.**9**(1999), 671-681. MR**2001f:30024****[N]**L. Nirenberg,*Remarks on strongly elliptic partial differential equations*, Commun. Pure Appl. Math.**8**(1955), 648-674. MR**17:742d****[NV]**F. Nazarov and A. Volberg,*Heating of the Beurling operator and estimates of its norms*, preprint.**[PV]**S. Petermichl and A. Volberg,*Heating of the Beurling operator*:*Weakly quasiregular mappings are quasiregular*, preprint.**[R]**S. Rickman,*Quasiregular Mappings*, Springer-Verlag, Berlin, 1993. MR**95g:30026****[R1]**Y. G. Reshetnyak,*Stability theorems for mappings with bounded distortion*, Seberian Math. J.**9**(1968), 499-512. MR**37:6459****[R2]**-,*Space Mappings with Bounded Distortion*, Transl. Math. Monographs, vol. 73, Amer. Math. Soc., Providence, RI, 1989. MR**90d:30067****[S1]**V. Sverák,*Examples of rank-one convex functions*, Proc. Roy. Soc. Edinburgh**114A**(1990), 237-242. MR**91h:26012****[S2]**-,*Rank-one convexity does not imply quasiconvexity*, Proc. Roy. Soc. Edinburgh**120A**(1992), 185-189. MR**93b:49026****[S3]**-,*New examples of quasiconvex functions*, Arch. Rational Mech. Anal.**119**(1992), 293-300. MR**93h:90072****[V]**M. Vuorinen, editor,*Quasiconformal Space Mappings*, Lecture Notes in Math., vol. 1508, Springer-Verlag, 1992. MR**94c:30030****[Y1]**B. Yan,*Remarks on**-stability of the conformal set in higher dimensions*, Ann. Inst. H. Poincaré, Analyse Non Linéaire**13(6)**(1996), 691-705. MR**97i:49043****[Y2]**-,*On rank-one convex and polyconvex conformal energy functions with slow growth*, Proc. Roy. Soc. Edinburgh**127A**(1997), 651-663. MR**98c:73033****[YZ]**B. Yan and Z. Zhou,*Stability of weakly almost conformal mappings*, Proc. AMS**126**(1998), 481-489. MR**98d:35064**

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Additional Information

**Tadeusz Iwaniec**

Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244

Email:
tiwaniec@mailbox.syr.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02914-8

Keywords:
Jacobians,
sharp estimates for singular integrals,
rank-one convexity,
quasiconformal mappings

Received by editor(s):
October 10, 1998

Published electronically:
January 8, 2002

Additional Notes:
Supported in part by NSF grant DMS-9706611

Article copyright:
© Copyright 2002
American Mathematical Society