Boundary multifractal behaviour for harmonic functions in the ball
Résumé
It is well known that if $h$ is a nonnegative harmonic function in the ball of
$\RR^{d+1}$ or if $h$ is harmonic in the ball with integrable
boundary value, then the radial limit of $h$ exists at almost every point of the
boundary. In this paper, we are interested in the exceptional set of points
of divergence and in the speed of divergence at these points. In particular, we prove that
for generic harmonic functions and for any $\beta\in [0,d]$, the Hausdorff
dimension of the set of points
$\xi$ on the sphere such that
$h(r\xi)$ looks like $(1-r)^{-\beta}$ is equal to $d-\beta$.
Domaines
Analyse classique [math.CA]
Origine : Fichiers produits par l'(les) auteur(s)