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26. CJM 1998 (vol 50 pp. 40)

Engliš, Miroslav; Peetre, Jaak
Green's functions for powers of the invariant Laplacian
The aim of the present paper is the computation of Green's functions for the powers $\DDelta^m$ of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in $\CC^d$, we obtain a complete result for $m=1,2$ in all dimensions. For $m\ge3$ the formulas grow quite complicated so we restrict ourselves to the case of the unit disc ($d=1$) where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space $\Bbb P^d$ (for $d=1$, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.

Keywords:Invariant Laplacian, Green's functions, dilogarithm, trilogarithm, Legendre and Jacobi polynomials, hyperlogarithms
Categories:35C05, 33E30, 33C45, 34B27, 35J40

27. CJM 1997 (vol 49 pp. 1066)

Shibata, Tetsutaro
Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity
We consider the multiparameter nonlinear Sturm-Liouville problem $$\displaylines{ u''(x) - \sum_{k=1}^m\mu_k u(x)^{p_k} + \sum_{k=m+1}^n \mu_ku(x)^{p_k} = \lambda u(x)^q, \quad x \in I := (-1,1), \cr u(x) > 0, \quad x \in I, \cr u(-1) = u(1) = 0,\cr}$$ where $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n) \in \bar{R}_+^m \times R_+^{n-m} \bigl(R_+ := (0, \infty)\bigr)$ and $\lambda \in R$ are parameters. We assume that $$1 \le q \le p_1 < p_2 < \cdots < p_n < 2q + 3.$$ We shall establish an asymptotic formula of variational eigenvalue $\lambda = \lambda(\mu,\alpha)$ obtained by using Ljusternik-Schnirelman theory on general level set $N_{\mu,\alpha} (\alpha > 0:$ parameter of level set). Furthermore, we shall give the optimal condition of $\{(\mu, \alpha)\}$, under which $\mu_i (m + 1 \le i \le n: \hbox{\rm fixed})$ dominates the asymptotic behavior of $\lambda(\mu,\alpha)$.

Categories:34B15, 34B25

28. CJM 1997 (vol 49 pp. 583)

Pal, Janos; Schlomiuk, Dana
Summing up the dynamics of quadratic Hamiltonian systems with a center
In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection $I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$, $Q(x,y,z) = 0$ at a point $p$ of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

Categories:34C, 58F

29. CJM 1997 (vol 49 pp. 338)

Rousseau, C.; Toni, B.
Local bifurcations of critical periods in the reduced Kukles system
In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gr\"obner bases, theory of the resultant, Sturm's algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

Categories:34C25, 58F14

30. CJM 1997 (vol 49 pp. 212)

Coll, B.; Gasull, A.; Prohens, R.
Differential equations defined by the sum of two quasi-homogeneous vector fields
In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar\'e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Categories:34C05, 58F21
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