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Entire solutions of first-order
nonlinear partial differential equations


Author: Jill E. Hemmati
Journal: Proc. Amer. Math. Soc. 125 (1997), 1483-1485
MSC (1991): Primary 35F20
DOI: https://doi.org/10.1090/S0002-9939-97-03881-1
MathSciNet review: 1396979
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that any entire solution of an essentially nonlinear first-order partial differential equation in two variables must be linear.


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  • 1. S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551-558.
  • 2. L. Bers, Isolated singularities of minimal surfaces, Ann. of Math. (2) 53 (1951), 364-380. MR 13:244c
  • 3. F. John, Partial Differential Equations, 4th ed., Springer-Verlag, New York, 1982. MR 80f:35001 (3rd ed.)
  • 4. K. Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2=1$, Math. Ann. 127 (1954), 130-134. MR 15:961e
  • 5. Dmitry Khavinson, A note on entire solutions of the eiconal [eikonal] equation, Amer. Math. Monthly 102 (1995), no. 2, 159–161. MR 1315596, https://doi.org/10.2307/2975351
  • 6. Steven G. Krantz, Function theory of several complex variables, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 635928
  • 7. Gérard Letac and Jean Pradines, Seules les affinités préservent les lois normales, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 8, A399–A402 (French, with English summary). MR 0474460
  • 8. E. J. Mickle, A remark on a theorem of Serge Bernstein, Proc. Amer. Math. Soc. 1 (1950), 86-89. MR 12:13f
  • 9. J. C. C. Nitsche, Elementary proof of Bernstein's theorem on minimal surfaces, Ann. of Math. (2) 66 (1957), 593-594. MR 19:878f
  • 10. Johannes C. C. Nitsche, Lectures on minimal surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems; Translated from the German by Jerry M. Feinberg; With a German foreword. MR 1015936
  • 11. T. Rado, Zu einem Satze von S.Bernstein über Minimalflächen im Gromen, Math. Z. 26 (1927), 559-565.
  • 12. Orestes N. Stavroudis and Ronald C. Fronczek, Caustic surfaces and the structure of the geometrical image, J. Opt. Soc. Amer. 66 (1976), no. 8, 795–800. MR 0424014, https://doi.org/10.1364/JOSA.66.000795

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Additional Information

Jill E. Hemmati
Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701

DOI: https://doi.org/10.1090/S0002-9939-97-03881-1
Received by editor(s): November 28, 1995
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1997 American Mathematical Society