Nonlinear Cauchy-Riemann operators in $\mathbb {R}^{n}$
HTML articles powered by AMS MathViewer
- by Tadeusz Iwaniec PDF
- Trans. Amer. Math. Soc. 354 (2002), 1961-1995 Request permission
Abstract:
This paper has arisen from an effort to provide a comprehensive and unifying development of the $L^{p}$-theory of quasiconformal mappings in $\mathbb {R}^{n}$. 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 $\mathcal {D}^{-}$ and $\mathcal {D}^{+}$, which act on weakly differentiable deformations $f:\Omega \to \mathbb {R}^{n}$ of a domain $\Omega \subset \mathbb {R}^{n}$.
Solutions to the so-called Cauchy-Riemann equations $\mathcal {D}^{-}f=0$ and $\mathcal {D}^{+}f=0$ 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 $L^{p}$-estimate \begin{equation*}\|\mathcal {D}^{+}f\|_{p} \le A_{p}(n)\|\mathcal {D}^{-}f\|_{p}. \end{equation*}
In quest of the best constant $A_{p}(n)$, we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.
References
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Kari Astala, Distortion of area and dimension under quasiconformal mappings in the plane, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), no. 24, 11958–11959. MR 1251715, DOI 10.1073/pnas.90.24.11958
- Kari Astala, Planar quasiconformal mappings; deformations and interactions, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) Springer, New York, 1998, pp. 33–54. MR 1488445
- Kari Astala, Analytic aspects of quasiconformality, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 617–626. MR 1648110
- Kari Astala, Tadeusz Iwaniec, and Eero Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27–56. MR 1815249, DOI 10.1215/S0012-7094-01-10713-8
- John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
- Rodrigo Bañuelos and Arthur Lindeman II, A martingale study of the Beurling-Ahlfors transform in $\textbf {R}^n$, J. Funct. Anal. 145 (1997), no. 1, 224–265. MR 1442167, DOI 10.1006/jfan.1996.3022
- R. Bañuelos and P. J. Méndez-Hernández, Space-time Brownian motion and the Beurling-Ahlfors transform, preprint (2001).
- Albert Baernstein II and Stephen J. Montgomery-Smith, Some conjectures about integral means of $\partial f$ and $\overline \partial f$, Complex analysis and differential equations (Uppsala, 1997) Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 64, Uppsala Univ., Uppsala, 1999, pp. 92–109. MR 1758918
- Rodrigo Bañuelos and Gang Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575–600. MR 1370109, DOI 10.1215/S0012-7094-95-08020-X
- Donald L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158 (1988), 75–94. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976214
- Donald L. Burkholder, A proof of Pełczynśki’s conjecture for the Haar system, Studia Math. 91 (1988), no. 1, 79–83. MR 957287, DOI 10.4064/sm-91-1-79-83
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890, DOI 10.1007/978-3-642-51440-1
- S. K. Donaldson and D. P. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), no. 3-4, 181–252. MR 1032074, DOI 10.1007/BF02392736
- F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
- L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 1, 17–35 (English, with English and French summaries). MR 1259100, DOI 10.1016/S0294-1449(16)30194-9
- T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), no. 6, 1–16 (English, with German and Russian summaries). MR 719167, DOI 10.4171/ZAA/37
- Tadeusz Iwaniec, Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators, Studia Math. 75 (1983), no. 3, 293–312. MR 722254, DOI 10.4064/sm-75-3-293-312
- Tadeusz Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136 (1992), no. 3, 589–624. MR 1189867, DOI 10.2307/2946602
- —, Integrability theory of the Jacobians, Lipschitz Lectures in Bonn, preprint 36, Sonderforschungsbereich 256, Bonn, 1995.
- —, On the concept of the weak Jacobian and Hessian, Report of the University of Jyväskylä No 83, dedicated to Olli Martio (2001), 181-205.
- Tadeusz Iwaniec and Adam Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. 125 (1993), no. 1, 25–79. MR 1241286, DOI 10.1007/BF00411477
- Tadeusz Iwaniec and Gaven Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), no. 1, 29–81. MR 1208562, DOI 10.1007/BF02392454
- Tadeusz Iwaniec and Gaven Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25–57. MR 1390681
- T. Iwaniec and G. Martin, Geometric Function Theory and Nonlinear Analysis, Oxford Mathematical Monographs, 2001.
- T. Iwaniec, L. Migliaccio, L. Nania, and C. Sbordone, Integrability and removability results for quasiregular mappings in high dimensions, Math. Scand. 75 (1994), no. 2, 263–279. MR 1319735, DOI 10.7146/math.scand.a-12519
- T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143–161. MR 1288682, DOI 10.1515/crll.1994.454.143
- A. Nicolau and J. Orobitg, Joint approximation in BMO, J. Funct. Anal. 173 (2000), no. 1, 21–48. MR 1760276, DOI 10.1006/jfan.1999.3552
- T. Iwaniec, G. Verchota and A. Vogel, The failure of rank-one connections, to appear in Arch. Rat. Mech. Anal.
- Juan J. Manfredi, Quasiregular mappings from the multilinear point of view, Fall School in Analysis (Jyväskylä, 1994) Report, vol. 68, Univ. Jyväskylä, Jyväskylä, 1995, pp. 55–94. MR 1351044
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1
- Stefan Müller, Vladimir Šverák, and Baisheng Yan, Sharp stability results for almost conformal maps in even dimensions, J. Geom. Anal. 9 (1999), no. 4, 671–681. MR 1757584, DOI 10.1007/BF02921978
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- F. Nazarov and A. Volberg, Heating of the Beurling operator and estimates of its norms, preprint.
- S. Petermichl and A. Volberg, Heating of the Beurling operator: Weakly quasiregular mappings are quasiregular, preprint.
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- Ju. G. Rešetnjak, Stability theorems for mappings with bounded distortion, Sibirsk. Mat. Ž. 9 (1968), 667–684 (Russian). MR 0230901
- Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644, DOI 10.1090/mmono/073
- Vladimír Šverák, Examples of rank-one convex functions, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), no. 3-4, 237–242. MR 1055547, DOI 10.1017/S0308210500024410
- Vladimír Šverák, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 185–189. MR 1149994, DOI 10.1017/S0308210500015080
- Vladimír Šverák, New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119 (1992), no. 4, 293–300. MR 1179688, DOI 10.1007/BF01837111
- Matti Vuorinen (ed.), Quasiconformal space mappings, Lecture Notes in Mathematics, vol. 1508, Springer-Verlag, Berlin, 1992. A collection of surveys 1960–1990. MR 1187085, DOI 10.1007/BFb0094234
- Baisheng Yan, Remarks on $W^{1,p}$-stability of the conformal set in higher dimensions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 6, 691–705 (English, with English and French summaries). MR 1420494, DOI 10.1016/S0294-1449(16)30119-6
- Baisheng Yan, On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 3, 651–663. MR 1453286, DOI 10.1017/S0308210500029954
- Baisheng Yan and Zhengfang Zhou, Stability of weakly almost conformal mappings, Proc. Amer. Math. Soc. 126 (1998), no. 2, 481–489. MR 1415344, DOI 10.1090/S0002-9939-98-04079-9
Additional Information
- Tadeusz Iwaniec
- Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
- Email: tiwaniec@mailbox.syr.edu
- Received by editor(s): October 10, 1998
- Published electronically: January 8, 2002
- Additional Notes: Supported in part by NSF grant DMS-9706611
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1881026