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Entire solutions of certain partial differential equations and factorization of partial derivatives
Author:
Bao Qin Li
Journal:
Trans. Amer. Math. Soc. 357 (2005), 3169-3177
MSC (2000):
Primary 32A15, 32A22, 35F20
Posted:
October 28, 2004
MathSciNet review:
2135740
Full-text PDF Free Access
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Abstract: We show that the problem of characterizing entire solutions of certain partial differential equations and the problem of characterizing common right factors of partial derivatives of meromorphic functions in  are closely related, and characterizations will be given using their relations.
- 1.
Carlos
A. Berenstein and Roger
Gay, Complex variables, Graduate Texts in Mathematics,
vol. 125, Springer-Verlag, New York, 1991. An introduction. MR 1107514
(92f:30001)
- 2.
H. Cartan, Sur les zeros des combinaisons linéaires de
 functions holomorphes données, Mathematica (Cluf) 7 (1933), 5-31.
- 3.
R.
Courant and D.
Hilbert, Methods of mathematical physics. Vol. II, Wiley
Classics Library, John Wiley & Sons Inc., New York, 1989. Partial
differential equations; Reprint of the 1962 original; A Wiley-Interscience
Publication. MR
1013360 (90k:35001)
- 4.
Der-Chen
Chang, Bao
Qin Li, and Chung-Chun
Yang, On composition of meromorphic functions in several complex
variables, Forum Math. 7 (1995), no. 1,
77–94. MR
1307956 (95i:32004), http://dx.doi.org/10.1515/form.1995.7.77
- 5.
P.
R. Garabedian, Partial differential equations, John Wiley
& Sons Inc., New York, 1964. MR 0162045
(28 #5247)
- 6.
Fred
Gross, Factorization of meromorphic functions, Mathematics
Research Center, Naval Research Laboratory, Washington, D. C., 1972. MR 0407251
(53 #11030)
- 7.
Jill
E. Hemmati, Entire solutions of first-order
nonlinear partial differential equations, Proc.
Amer. Math. Soc. 125 (1997), no. 5, 1483–1485. MR 1396979
(97g:35022), http://dx.doi.org/10.1090/S0002-9939-97-03881-1
- 8.
A.
V. Jategaonkar, Elementary proof of a theorem of P. Montel on
entire functions, J. London Math. Soc. 40 (1965),
166–170. MR 0170007
(30 #248)
- 9.
Fritz
John, Partial differential equations, 4th ed., Applied
Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655
(87g:35002)
- 10.
Dmitry
Khavinson, A note on entire solutions of the eiconal [eikonal]
equation, Amer. Math. Monthly 102 (1995), no. 2,
159–161. MR 1315596
(95j:35132), http://dx.doi.org/10.2307/2975351
- 11.
Bao
Qin Li and Chung-Chun
Yang, Factorization of meromorphic functions in several complex
variables, Several complex variables in China, Contemp. Math.,
vol. 142, Amer. Math. Soc., Providence, RI, 1993,
pp. 61–74. MR 1208784
(94a:32006), http://dx.doi.org/10.1090/conm/142/1208784
- 12.
A.
I. Markushevich, Entire functions, Translated from the Russian
by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, American
Elsevier Publishing Co., Inc., New York, 1966. MR 0199395
(33 #7541b)
- 13.
P. Montel, Lecons sur les families normales de functions analytique et leurs applications, Gauthier-Villars, Paris, 1927.
- 14.
Jeffrey
Rauch, Partial differential equations, Graduate Texts in
Mathematics, vol. 128, Springer-Verlag, New York, 1991. MR 1223093
(94e:35002)
- 15.
Elias
G. Saleeby, Entire and meromorphic solutions of Fermat type partial
differential equations, Analysis (Munich) 19 (1999),
no. 4, 369–376. MR 1743529
(2001c:35050)
- 16.
B.
V. Shabat, Introduction to complex analysis. Part II,
Translations of Mathematical Monographs, vol. 110, American
Mathematical Society, Providence, RI, 1992. Functions of several variables;
Translated from the third (1985) Russian edition by J. S. Joel. MR 1192135
(93g:32001)
- 17.
Wilhelm
Stoll, Introduction to value distribution theory of meromorphic
maps, Complex analysis (Trieste, 1980) Lecture Notes in Math.,
vol. 950, Springer, Berlin, 1982, pp. 210–359. MR 672787
(84a:32041)
- 18.
Al
Vitter, The lemma of the logarithmic derivative in several complex
variables, Duke Math. J. 44 (1977), no. 1,
89–104. MR
0432924 (55 #5903)
- 1.
- C.A. Berenstein and R. Gay, Complex Variables, Springer-Verlag, New York, 1991. MR 1107514 (92f:30001)
- 2.
- H. Cartan, Sur les zeros des combinaisons linéaires de functions holomorphes données, Mathematica (Cluf) 7 (1933), 5-31.
- 3.
- R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol II, partial differential equations, Interscience, New York, 1962. MR 1013360 (90k:35001)
- 4.
- D.C. Chang, B.Q. Li, and C.C. Yang, On composition of meromorphic functions in several complex variables, Forum Math 7 (1995), 77-94. MR 1307956 (95i:32004)
- 5.
- P.R. Garabedian, Partial Differential Equations, Wiley, New York, 1964. MR 0162045 (28:5247)
- 6.
- F. Gross, Factorization of Meromorphic Functions, U.S. Government Printing Office, Washington, D.C., 1972. MR 0407251 (53:11030)
- 7.
- J. Hemmati, Entire solutions of first-order nonlinear partial differential equations, Proc. Amer. Math. Soc. 125 (1997), 1483-1485. MR 1396979 (97g:35022)
- 8.
- A.V. Jategaonkar, Elementary proof of a theorem of P. Motel on entire functions, J. London Math. Soc. 40 (1965), 166-170. MR 0170007 (30:248)
- 9.
- F. John, Partial Differential Equations, Springer-Verlag, New York, 1982. MR 0831655 (87g:35002)
- 10.
- D. Khavinson, A note on entire solutions of the eiconal equation, Amer. Math. Mon. 102 (1995), 159-161. MR 1315596 (95j:35132)
- 11.
- B.Q. Li and C.C. Yang, Factorization of meromorphic functions in several complex variables, Contemporary Mathematics 142 (1993), 61-74. MR 1208784 (94a:32006)
- 12.
- J. Markushevich, Entire Functions, Amer. Elsevier Pub. Co., New York, 1966. MR 0199395 (33:7541b)
- 13.
- P. Montel, Lecons sur les families normales de functions analytique et leurs applications, Gauthier-Villars, Paris, 1927.
- 14.
- J. Rauch, Partial Differential Equations, Springer-Verlag, New York, 1991. MR 1223093 (94e:35002)
- 15.
- E.G. Saleeby, Entire and meromorphic solutions of Fermat type partial differential equations, Analysis 19 (1999), 369-376. MR 1743529 (2001c:35050)
- 16.
- B.V. Shabat, Introduction to Complex Analysis, part II, Functions of several variables, Translation Mathematical Monographs, Vol. 110, American Mathematical Society, Providence, RI, 1992. MR 1192135 (93g:32001)
- 17.
- W. Stoll, Introduction to the Value Distribution Theory of Meromorphic Functions, Springer-Verlag, New York, 1982. MR 0672787 (84a:32041)
- 18.
- A. Vitter, The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), 89-104. MR 0432924 (55:5903)
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Additional Information
Bao Qin Li
Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199
Email:
libaoqin@fiu.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03745-6
PII:
S 0002-9947(04)03745-6
Received by editor(s):
October 16, 2003
Posted:
October 28, 2004
Additional Notes:
The author was supported in part by National Science Foundation Grant DMS-0100486
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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