Solutions d'équations elliptiques semi-linéaires singulières

Résumé Récemment T.Kusano et C. A.Swanson ont étudié l'équation elliptique semilinéaire u+j(x)u ^– =0,x R ⁿ,n 3et 0Ils ont montré l'existence d'une solution positive globale se comportant comme ¦x¦ ^2–n pour ¦x¦ grand. Nous démontrons l'existence et l'unicité d'une telle solution pour >/ 0.

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Summary T.Kusano and C. A.Swanson have recently studied the semilinear elliptic equationu+j(x)u ^– =0,x R ⁿ,n3 and 0< < 1. They have shown the existence of a global positive solution behaving like ¦x¦^2–n for large ¦x¦. We establish the existence and uniqueness of such a solution when > 0.

     

Sous-solutions d'équations elliptiques dans L1

We study subsolutions for semilinear elliptic boundary value problems in L1. We consider as well nonlinear as linear boundary conditions. The nonlinear functions may be multivated. We characterize in terms of p.d.e. the subsolutions defined by a nonlinear functional analysis argument. Applications are given to obtain existence results for semilinear elliptic boundary value problems and comparison and estimates for nonlinear parabolic boundary value problems.     

Solutions périodiques des équations d'évolution

We study here the necessary and sufficient condition for the existence of periodic solutions for evolution equations in the case of linear unbounded maximal monotonous and symmetric operators. We present also an estimate of the convergence time to the periodic states in the finite-dimensional case.     

Sur les fonctions arithmétiques non entières

A theorem of Baker says that a function F entire on ? ^d such that F(? ^d ) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain

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, j = 1,..., d}.

The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane

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.

In this article, we show how this characteristic function can also be used to bound the growth of multivariate functions defined on

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that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.

     

Sur un théorème de M. Wiman dans la théorie des fonctions entières

Sans résuméMes sentiments de profonde reconnaissance vont à M. le Prof. E. Landau pour m'avoir suggéré le sujet du présent travail et pour la cordialité de son acceuil durant mon séjour à Göttingen.     

Solutions faibles d'énergie infinie pour les équations de Navier—Stokes dans ℝ3

We prove existence of suitable weak solutions for the Navier-Stokes equations on (0, ∞) × ℝ³ when the initial value is uniformly locally square integrable and vanishes at infinity.     

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.     

Fonctions entières et intégrales de Fourier multiples

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Fonctions entières et intégrales de fourier multiples

Commentarii Mathematici Helvetici -     

Sur une propriété des séries entières

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Sur un certain système d'équations et la recherche des figures d'équilibre

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