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Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates


Authors: Adam Coffman, Yifei Pan and Yuan Zhang
Journal: Trans. Amer. Math. Soc. 369 (2017), 4865-4887
MSC (2010): Primary 35J46; Secondary 30G20, 32Q65
DOI: https://doi.org/10.1090/tran/6845
Published electronically: February 13, 2017
MathSciNet review: 3632553
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish elliptic regularity for nonlinear, inhomogeneous
Cauchy-Riemann equations under weak assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.


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  • [AIM] Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875
  • [BBC] Laurent Baratchart, Alexander Borichev, and Slah Chaabi, Pseudo-holomorphic functions at the critical exponent, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 9, 1919-1960. MR 3531666, https://doi.org/10.4171/JEMS/634
  • [CP] Adam Coffman and Yifei Pan, Smooth counterexamples to strong unique continuation for a Beltrami system in $ \mathbb{C}^2$, Comm. Partial Differential Equations 37 (2012), no. 12, 2228-2244. MR 3005542, https://doi.org/10.1080/03605302.2012.668259
  • [C$_1$] Paul J. Cohen, Topics in the Theory of Uniqueness of Trigonometrical Series, ProQuest LLC, Ann Arbor, MI, 1958. Thesis (Ph.D.)-The University of Chicago. MR 2611474
  • [C$_2$] Paul J. Cohen, On Green's theorem, Proc. Amer. Math. Soc. 10 (1959), 109-112. MR 0104249
  • [Conway] John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
  • [CV] Julià Cufí and Joan Verdera, A general form of Green's formula and the Cauchy integral theorem, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2091-2102. MR 3314118, https://doi.org/10.1090/S0002-9939-2014-12418-X
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • [GR] Xianghong Gong and Jean-Pierre Rosay, Differential inequalities of continuous functions and removing singularities of Rado type for $ J$-holomorphic maps, Math. Scand. 101 (2007), no. 2, 293-319. MR 2379291
  • [GM] J. D. Gray and S. A. Morris, When is a function that satisfies the Cauchy-Riemann equations analytic?, Amer. Math. Monthly 85 (1978), no. 4, 246-256. MR 0470179
  • [H] Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382
  • [IS$_1$] S. Ivashkovich and V. Shevchishin, Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math. 136 (1999), no. 3, 571-602. MR 1695206, https://doi.org/10.1007/s002220050319
  • [IS$_2$] S. Ivashkovich and V. Shevchishin, Local properties of $ J$-complex curves in Lipschitz-continuous structures, Math. Z. 268 (2011), no. 3-4, 1159-1210. MR 2818746, https://doi.org/10.1007/s00209-010-0713-6
  • [M] Charles B. Morrey Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior, Amer. J. Math. 80 (1958), 198-218. MR 0106336
  • [N] Raghavan Narasimhan, Complex analysis in one variable, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781130
  • [R] Jean-Pierre Rosay, Uniqueness in rough almost complex structures, and differential inequalities, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 6, 2261-2273 (English, with English and French summaries). MR 2791657
  • [ST] Bernd Siebert and Gang Tian, Lectures on pseudo-holomorphic curves and the symplectic isotopy problem, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., vol. 1938, Springer, Berlin, 2008, pp. 269-341. MR 2463700, https://doi.org/10.1007/978-3-540-78279-7_5
  • [S] Jean-Claude Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 165-189. MR 1274929
  • [T] Clifford H. Taubes, $ {\rm SW}\Rightarrow {\rm Gr}$: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), no. 3, 845-918. MR 1362874, https://doi.org/10.1090/S0894-0347-96-00211-1

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Additional Information

Adam Coffman
Affiliation: Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseum Boulevard, Fort Wayne, Indiana 46805-1499
Email: CoffmanA@ipfw.edu

Yifei Pan
Affiliation: College of Mathematics and Information Sciences, Jiangxi Normal University, Nanchang, People’s Republic of China
Email: Pan@ipfw.edu

Yuan Zhang
Affiliation: Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseum Boulevard, Fort Wayne, Indiana 46805-1499
Email: ZhangYu@ipfw.edu

DOI: https://doi.org/10.1090/tran/6845
Received by editor(s): February 26, 2015
Received by editor(s) in revised form: March 2, 2015, and July 27, 2015
Published electronically: February 13, 2017
Additional Notes: The first author is the corresponding author
This paper was presented to the American Mathematical Society at the Spring 2015 Central Sectional Meeting in East Lansing, Michigan
Article copyright: © Copyright 2017 American Mathematical Society

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