Description of Entire Solutions of Eiconal Type Equations

Chang, Der-Chen ; Li, Bao Qin

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 249-259

Résumé

The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations ${{({{X}_{1}}(u))}^{2}}\,+\,{{({{X}_{2}}(u))}^{2}}\,=\,1$ as special cases, where ${{X}_{1}}\,=\,{{p}_{1}}\partial /\partial {{z}_{1}}\,+\,{{p}_{2}}\partial /\partial {{z}_{2}},\,{{X}_{2}}\,=\,{{p}_{3}}\partial /\partial {{z}_{1}}\,+\,{{p}_{4}}\partial /\partial {{z}_{2}}$ are linearly independent operators with ${{p}_{j}}$ being arbitrary polynomials in ${{\mathbf{C}}^{2}}$ .

Détail

DOI : 10.4153/CMB-2011-080-8

Mots-clés : 32A15, 35F20, entire solution, eiconal equation, polynomial, transcendental function

Bibliographie

[1] [1] Bernstein, S., Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26(1927), no. 1, 551–558. Google Scholar | DOI

[2] [2] Calin, O. and Chang, D. C., Geometric Mechanics on Riemannian Manifolds. Applications to Partial Differential Equations. Birkhäuser, Boston, 2005. Google Scholar

[3] [3] Courant, R. and Hilbert, D., Methods of Mathematical Physics. II. Partial Differential Equations. Interscience John Wiley & Sons, New York, 1989. Google Scholar

[4] [4] Chang, D. C., Li, B. Q., and Yang, C. C., On composition of meromorphic functions in several complex variables. Forum Math 7(1995), no. 1, 77–94. Google Scholar | DOI

[5] [5] Garabedian, P. R., Partial Differential Equations. John Wiley, New York, 1964. Google Scholar

[6] [6] Jörgens, K., Über die Lösungen der Differentialgeichung rt – s2 = 1. Math. Ann. 127(1954), 130–134. Google Scholar | DOI

[7] [7] Khavinson, D., A note on entire solutions of the eiconal equation. Amer. Math. Monthly 102(1995), no. 2, 159–161. Google Scholar | DOI

[8] [8] Li, B. Q., Entire solutions of eiconal type equations. Arch. Math. 89(2007), no. 4, 350–357. Google Scholar

[9] [9] Nitsche, J. C. C., Elementary proof of Bernstein's theorem on minimal surfaces. Ann. of Math. 66(1957), 593–594. Google Scholar | DOI

[10] [10] Saleeby, E. G., Entire and meromorphic solutions of Fermat type partial differential equations. Analysis 19(1999), no. 4, 369–376. Google Scholar

[11] [11] Stoll, W., Introduction to the value distribution theory of meromorphic functions. In: Complex Analysis. Lecture Notes in Math. 950. Springer-Verlag, Berlin, 1982. Google Scholar

[12] [12] Vitter, A., The lemma of the logarithmic derivative in several complex variables. Duke Math. J. 44(1977), no. 1, 89–104. Google Scholar | DOI