Une construction de fonctions plurisousharmoniques nous permet, en utilisant les techniques de Hörmander, d’obtenir un résultat de $d^{\text{'}\text{'}}$-cohomologie à croissance. Les méthodes de B. Malgrange nous fournissent alors deux applications aux systèmes différentiels à coefficients constants.

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

Si $B$ est une boule ouverte contenue dans le domaine euclidien $\Omega$, tout filtre sur $B$, tendant non tangentiellement vers un point de $\partial \Omega \cap \partial B$, converge vers un point minimal dans le compactifié de Martin de $\Omega$. On donne une application, et une variante dans le cas plan, et on termine par un contre-exemple apportant une solution négative à un problème de R.S. Martin. L’idée générale de l’article est d’établir des variantes des inégalités de Harnack pour déterminer la frontière de Martin du domaine.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

We give a characterization of functions that are uniformly approximable on a compact subset $K$ of ${ℝ}^n$ by biharmonic functions in neighborhoods of $K$.

Let $\Delta u={\Sigma }_i\frac{{\partial }^2}{{\partial }_{x_i}^2}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^2}{\partial x_i\partial x_j}u+{\Sigma }_i{b}_i\frac{\partial }{\partial x_i}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^n$ of class $C^{1,1}$. There is a constant $c\>0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that$$c^{-1}{G}_{\Delta }^{\Omega }\le G_L^{\Omega }\le c{G}_{\Delta }^{\Omega }\text{on}\Omega \times \Omega ,$$where ${\gamma }_{\Delta }^{\Omega }$ (resp. $G_L^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.