Aspects des fonctions elliptiques. \\ Solutions périodiques d'équations différentielles.\\ Métriques pseudo-cylindriques. \\ Problèmes isopérimètriques plans

Abstract : \begin(center) (\bf ABSTRACT) \end(center) The summary of the research works that one proposes to present itself compose four parts \footnote (The references are those of the list of my publications.) * In the first, one examines properties of the Weierstrass elliptic function \ $\wp(z,\omega,\omega')$\ and the classical theta functions \ $\theta(v,\tau),$\ in light of a new trigonometric development. One determines the coefficients explicitly as a function of \ $\tau$, that are in fact hypergeometric functions. (\it [P1], [P6]).\\ In particular, it permits a new construction of the theory of elliptic functions and to recover some properties. Notably, the one concerning the function zeta of Jacobi \ $Zn(z,k).$\ Also, one puts in evidence of modular relations between these coefficients. \ (\it [P7], [P19], [P22]).\\ * In the second part, one is interested in periodic solutions of some ordinary differential equations. More precisely we look in for the growth of the function period depending on the energy and to sufficient conditions of its monotony. We give new criteria implying the period function increases. (\it [P5]). One shows in particular that all the known criteria - those of S.N. Chow, R. Schaaf, C. Chicone and F. Rothe - are not optimal. \\ One is also interested in the monotony of the period of the system of Liénard with a center at the origine. (\it [P8]). On gives in particular a simpler proof of Christopher's result and Devlin . . (\it [T30] ) While using trigonometric sets, one shows the existence of periodic solutions for an perturbed equation of Duffing type. (\it [P10], [P12]). \\ Finally, one uses with success a method of M. Farkas concerning the controllability of the period to show the existence of a periodic solution of the perturbed Liénard equation. (\it [P13] ).\\ * In the third part, we put in evidence properties of Riemannian and Ricci curvature for some metrics with positive constant curvature scalar, as well as their singularities. (\it [P3], [P4]).\\ These metrics in finished number are called pseudo-cylindrical. Besides, they have a harmonic curvature and a non parallel Ricci curvature, and are solutions of the singular Yamabe problem on the standard sphere punctured of two points \ $S^n - $$p_1,p_2$$.$\ One also examines their asymptotic properties. Moreover, for some values of \ $n \ = 3,4$\ or \ $6$\ one can determine these metrics explicitly. One is also interested in the problem of existence of warped metrics of A. Derdzinski. (\it [P11], [P15]).\\ * Finally, in the last part, one considers the isoperimetric inequalities of Bonnesen type in relation with conjectures of P. Lévy and X.M. Zhang on plane polygons. In particular, we propose a more general conjecture . (\it [P2], [P9], [P18]).
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https://tel.archives-ouvertes.fr/tel-00003633
Contributor : Raouf Chouikha <>
Submitted on : Thursday, October 23, 2003 - 6:00:05 PM
Last modification on : Tuesday, February 5, 2019 - 11:44:10 AM
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Raouf Chouikha. Aspects des fonctions elliptiques. \\ Solutions périodiques d'équations différentielles.\\ Métriques pseudo-cylindriques. \\ Problèmes isopérimètriques plans. Mathématiques [math]. Université de Rouen, 2003. ⟨tel-00003633⟩

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