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Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball


Authors: J. Berkovits and J. Mawhin
Journal: Trans. Amer. Math. Soc. 353 (2001), 5041-5055
MSC (1991): Primary 35B10, 35L05, 35L20, 11J70, 47H10
Published electronically: July 13, 2001
MathSciNet review: 1852093
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Abstract:

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on $[0,T] \times B^n[a]$ for the equation

\begin{displaymath}u_{tt} - \Delta u = f(t,\vert x\vert,u),\end{displaymath}

where $\Delta$ is the classical Laplacian operator, and $B^n[a]$ denotes the open ball of center $0$ and radius $a$ in ${\mathbb R}^n.$ When $\alpha = a/T$ is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that $0$ is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.


References [Enhancements On Off] (What's this?)

  • 1. Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, Inc., New York, 1966. MR 0208797
  • 2. A. Baker, Continued fractions of transcendental numbers, Mathematika 9 (1962), 1–8. MR 0144853
  • 3. A. K. Ben-Naoum and J. Berkovits, On the existence of periodic solutions for semilinear wave equation on a ball in 𝐑ⁿ with the space dimension 𝐧 odd, Nonlinear Anal. 24 (1995), no. 2, 241–250. MR 1312593, 10.1016/0362-546X(94)E0069-S
  • 4. A. B. Mikhaĭlov and V. P. Podporin, Determining the domain of analyticity of solutions of a Cauchy problem, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk. 1-2 (1992), 20–24, 89 (Russian, with Russian summary). MR 1186306
  • 5. A. K. Ben-Naoum and J. Mawhin, Periodic solutions of some semilinear wave equations on balls and on spheres, Topol. Methods Nonlinear Anal. 1 (1993), no. 1, 113–137. MR 1215261
  • 6. F. Bernstein, Ueber eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem, Math. Ann. 71 (1912), 417-439.
  • 7. E. Borel, Sur les équations aux dérivées partielles à coefficients constants et les fonctions non analytiques, C.R. Acad. Sci. Paris 121 (1895), 933-935.
  • 8. E. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247-271.
  • 9. E. Borel, Sur un problème de probabilités relatifs aux fractions continues, Math. Ann. 72 (1912), 578-584.
  • 10. Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R. I., 1976, pp. 1–308. MR 0405188
  • 11. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
  • 12. V. Jarník, Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz. 36 (1928), 91-106.
  • 13. Serge Lang, Introduction to Diophantine approximations, 2nd ed., Springer-Verlag, New York, 1995. MR 1348400
  • 14. Boris Rybak, Paul Janssens, and M. Jessel (eds.), Mélanges, Presses Universitaires de Bruxelles, Brussels, 1978 (French). Offert à Th. Vogel, à l’occasion de son 75ème anniversaire, par ses collègues et amis. [Papers offered to Th. Vogel on his 75th birthday by his colleagues and friends]. MR 825534
  • 15. Jean Mawhin, Semilinear equations of gradient type in Hilbert spaces and applications to differential equations, Nonlinear differential equations (Proc. Internat. Conf., Trento, 1980), Academic Press, New York-London, 1981, pp. 269–282. MR 615106
  • 16. Jean Mawhin, Continuation theorems and periodic solutions of ordinary differential equations, Topological methods in differential equations and inclusions (Montreal, PQ, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472, Kluwer Acad. Publ., Dordrecht, 1995, pp. 291–375. MR 1368675
  • 17. P. J. McKenna, On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational, Proc. Amer. Math. Soc. 93 (1985), no. 1, 59–64. MR 766527, 10.1090/S0002-9939-1985-0766527-X
  • 18. Ivan Niven and Herbert S. Zuckerman, An introduction to the theory of numbers, 4th ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 572268
  • 19. Andrew M. Rockett and Peter Szüsz, Continued fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1188878
  • 20. George R. Sell, The prodigal integral, Amer. Math. Monthly 84 (1977), no. 3, 162–167. MR 0427556
  • 21. Jeffrey Shallit, Real numbers with bounded partial quotients: a survey, Enseign. Math. (2) 38 (1992), no. 1-2, 151–187. MR 1175525
  • 22. M. W. Smiley, Time-periodic solutions of nonlinear wave equations in balls, Oscillations, bifurcation and chaos (Toronto, Ont., 1986) CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 1987, pp. 287–297. MR 909918
  • 23. G. Watson, A Treatise on the Theory of Bessel Functions, University Press, Cambridge, 1922.

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Additional Information

J. Berkovits
Affiliation: Department of Mathematics, University of Oulu, Oulu, Finland
Email: jberkovi@sun3.oulu.fi

J. Mawhin
Affiliation: Université Catholique de Louvain, Institut Mathématique, B-1348 Louvain-la-Neuve, Belgium
Email: mawhin@amm.ucl.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02875-6
Keywords: Diophantine approximations, Bessel functions, wave equation, periodic solutions
Received by editor(s): December 28, 1999
Published electronically: July 13, 2001
Article copyright: © Copyright 2001 American Mathematical Society