Entire solutions of first-order nonlinear partial differential equations. (English) Zbl 0870.35024
Summary: We show that any entire solution of a nonlinear first-order partial differential equation of the form \(F(u_x,u_y)= 0\), where \(F\) is an entire function and the zero set of \(F\) does not contain any complex lines, must be linear.
MSC:
| 35F20 | Nonlinear first-order PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
References:
| [1] | S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551-558. · JFM 53.0670.01 |
| [2] | L. Bers, Isolated singularities of minimal surfaces, Ann. of Math. (2) 53 (1951), 364-380. · Zbl 0043.15901 |
| [3] | F. John, Partial Differential Equations, 4th ed., Springer-Verlag, New York, 1982. · Zbl 0472.35001 |
| [4] | K. Jörgens, Über die Lösungen der Differentialgleichung \(rt-s^2=1\), Math. Ann. 127 (1954), 130-134. · Zbl 0055.08404 |
| [5] | Dmitry Khavinson, A note on entire solutions of the eiconal [eikonal] equation, Amer. Math. Monthly 102 (1995), no. 2, 159 – 161. · Zbl 0845.35017 · doi: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. · Zbl 0471.32008 |
| [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). · Zbl 0372.60012 |
| [8] | E. J. Mickle, A remark on a theorem of Serge Bernstein, Proc. Amer. Math. Soc. 1 (1950), 86-89. · Zbl 0039.16902 |
| [9] | J. C. C. Nitsche, Elementary proof of Bernstein’s theorem on minimal surfaces, Ann. of Math. (2) 66 (1957), 593-594. · Zbl 0079.37702 |
| [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. · Zbl 0688.53001 |
| [11] | T. Rado, Zu einem Satze von S.Bernstein über Minimalflächen im Gromen, Math. Z. 26 (1927), 559-565. · JFM 53.0670.02 |
| [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. · doi:10.1364/JOSA.66.000795 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
