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Entire solutions of a variation of the eikonal equation and related PDEs

Part of: Holomorphic functions of several complex variables General first-order equations and systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  07 May 2020

Feng Lü
Feng Lü*
Affiliation:
College of Science, China University of Petroleum, Qingdao, Shandong266580, P.R. China ([email protected])
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Abstract

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

Keywords

entire solutionFermat functional equationcharacteristic methodpartial differential equation

MSC classification

Primary: 30D30: Meromorphic functions, general theory
Secondary: 32A15: Entire functions 35F20: Nonlinear first-order equations 32A22: Nevanlinna theory (local); growth estimates; other inequalities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 697 - 708
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

1.Baker, I. N., On a class of meromorphic functions, Proc. Amer. Math. Soc. 17 (1966), 819–822.CrossRefGoogle Scholar
2.Cartan, H., Sur les zéros des combinaisons linéaires de p fonctions holomorphes données, Mathematica (Cluf) 7 (1933), 5–31.Google Scholar
3.Courant, R. and Hilbert, D., Methods of mathematical physics, Partial Differential Equations, Volume II (Interscience, New York, 1962).Google Scholar
4.Garabedian, P. R., Partial differential equations (Wiley, New York, 1964).Google Scholar
5.Gross, F., On the equation $f^n+g^n=1$. I and II, Bull. Amer. Math. Soc. 72 (1966), 86–88; 74 (1968), 647–648.CrossRefGoogle Scholar
6.Hille, E., Ordinary differential equations in the complex domain (Dover, Mineola, NY, 1997).Google Scholar
7.Khavinson, D., A note on entire solutions of the eiconal equation, Amer. Math. Monthly 102 (1995), 159–161.CrossRefGoogle Scholar
8.Li, B. Q., Entire solutions of certain partial differential equations and factorization of partial derivatives, Trans. Amer. Math. Soc. 357 (2005), 3169–3177.CrossRefGoogle Scholar
9.Li, B. Q., Entire solutions of $(u_{z_1})^m+(u_{z_2})^n=e^g$, Nagoya Math. J. 178 (2005), 151–162.CrossRefGoogle Scholar
10.Li, B. Q., On certain functional and partial differential equations, Forum Math. 17 (2005), 77–86.CrossRefGoogle Scholar
11.Li, B. Q., On meromorphic solutions of $f^2 + g^2 = 1$, Math. Z. 258 (2008), 763–771.CrossRefGoogle Scholar
12.Li, B. Q., Fermat-type functional and partial differential equations, in The mathematical legacy of Leon Ehrenpreis, Springer Proceedings in Mathematics, Volume 16, pp. 209–222 (Springer, Milan, 2012).CrossRefGoogle Scholar
13.Li, B. Q., On meromorphic solutions of generalized Fermat equations, Int. J. Math. 25(1) (2014), 1450002.CrossRefGoogle Scholar
14.Li, B. Q., Estimates on derivatives and logarithmic derivatives of holomorphic functions and Picard's theorem, J. Math. Anal. Appl. 442 (2016), 446–450.CrossRefGoogle Scholar
15.Montel, P., leçons sur les families normales de functions analytiques et leurs applications (Gauthier-Villars, Paris, 1927).Google Scholar
16.Saleeby, E. G., On complex analytic solutions of certain trinomial functional and partial differential equations, Aequat. Math. 85 (2013), 553–562.10.1007/s00010-012-0154-xCrossRefGoogle Scholar
17.Stoll, W., Introduction to the value distribution theory of meromorphic functions (Springer-Verlag, New York, 1982).Google Scholar
18.Vitter, A., The lemma of the logarithmic derivative in seveal complex variables, Duke Math. J. 44 (1977), 89–104.CrossRefGoogle Scholar