Periodic solutions of the retarded Liénard equation

Consider the equation

Résolutions flasques des groupes réductifs connexes

A connected reductive group G over a (characteristic zero) field k may be written as a quotient $H/S$, where the k-group H is an extension of a quasitrivial torus by a simply connected group, and S is a flasque k-torus, central in H. Such presentations $G=H/S$ lead to a simplified approach to the Galois cohomology of G and related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recovers known formulas, in terms of S, for the quotient of the group $G(k)$ of rational points by R-equivalence, and for the Abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces. To cite this article: J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

étude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales

Sans résumé     

Tronquée solutions of the painlevé II equation

We study special solutions of the Painlevé II (PII) equation called tronquée solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a twodimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquée solutions of the PII equation. As an illustration, we consider the known Hastings-McLeod and Ablowitz-Segur solutions and some other solutions to show that they belong to the class of tronquée solutions and correspond to one or another type of singularity of the monodromy data.     

Boundedness and periodicity of solutions of the generalized liénard equation

Summary The object of this paper is to investigate, for large values of t, the solutions of Liénard type equations. A periodicity theorem is proved and several conditions which imply boundedness are discovered. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under contract No. AF 18(600)-1108.     

Croissance des Solutions des D-Modules algébriques holonomes – (Growth of solutions of holonomic algebraic D-modules)

We show that the cohomology of complexes of solutions of exponantial type associated to holonomic algebraic D-modules is constructible. We also compute the Euler–Poincaré index of such complexes.     

Une méthode des sur et sous solutions améliorée

The usual upper and lower solutions method for solving a second order nonlinear elliptic equation is iterative, with the drawback of a tricky, if not sometimes impossible, derivation of the (quite essential) uniform $C^2$ estimate, due to the occurence in the same equation of two successive iteration unknowns. We present here a method free of such a drawback, based on an elementary fixed point argument. To cite this article: P. Delanoë, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Non-radial least energy solutions of the generalized Hénon equation

In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in $|x|<a$ to that in $a<|x|<1$ is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric.     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

Comportement asymptotique des solutions d'équations elliptiques semi-linéaires dans RN

Summary We study the asymptotic behaviour in R^N of the solutions of the semilinear elliptic equation – u + u¦u¦^q-1=f where q > 1 and f is a function of L¹(R^N) with compact support. We obtain three rates of decay according to the value of q by respect to N/(N - 2) and we prove that the behaviour of u is isotropic when q(N+1)/(N– 1). We also give an asymptotic expansion of u in each case.     

Symétrie des grandes solutions d'équations elliptiques semi linéaires

Let g be a locally Lipschitz continuous function defined on $\mathbb{R}$. We assume that g satisfies the Keller–Osserman condition and there exists a positive real number a such that g is convex on $[a,\infty )$. Then any solution u of $?\mathrm{\Delta }u+g(u)=0$ in a ball B of ${\mathbb{R}}^N$, $N?2$, which tends to infinity on ?B, is spherically symmetric. To cite this article: A. Porretta, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation

For the boundary value problem