A positive measurable function K on a domain D in ${ℝ}^{n+1}$ is called a mean value density for temperatures if $u(0,0)=\int {\int }_{D}K(x,t)u(x,t)dxdt$ for all temperatures u on D̅. We construct such a density for some domains. The existence of a bounded density and a density which is bounded away from zero on D is also discussed.

Let $u$ be a $\delta$-subharmonic function with associated measure $\mu$, and let $v$ be a superharmonic function with associated measure $\nu$, on an open set $E$. For any closed ball $B(x,r)$, of centre $x$ and radius $r$, contained in $E$, let $ℳ(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\to 0$ with $0<s<t$ of the quotient $\left(ℳ\right(u,x,s)-ℳ(u,x,t\left)\right)/\left(ℳ\right(v,x,s)-ℳ(v,x,t\left)\right)$, lie between the upper and lower limits as $r\to 0+$ of the quotient $\mu \left(B\right(x,r\left)\right)/\nu \left(B\right(x,r\left)\right)$. This enables us to use some well-known measure-theoretic results to prove new variants and generalizations...

Soit $K$ un compact polynomialement convexe de ${\mathbf{C}}^n$ et $V_K$ son “potentiel logarithmique extrémal” dans ${\mathbf{C}}^n$. Supposons que $K$ est régulier (i.e. $V_K$ continue) et soit $f$ une fonction holomorphe sur un voisinage de $K$. On construit alors une suite $\{{\mathbf{P}}_{\ell }{\}}_{\ell \ge 1}$ de polynôme de $n$ variables complexes avec deg$({\mathbf{P}}_{)}\le \ell$ pour $\ell \ge 1$, telle que l’erreur d’approximation ${\max }_{z\in K}|f\left(z\right)-P_{\ell }\left(z\right)|$ soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de $f$ par rapport à $K$ et du degré de convergence $\ell$. Ce résultat est ensuite utilisé pour étendre...

The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results...

We study minimal thinness in the half-space $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d\&gt;0\}$ for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.