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Pré-Publication, Document De Travail Année : 2011

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 $d-\beta$.
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Dates et versions

hal-00635997 , version 1 (26-10-2011)
hal-00635997 , version 2 (22-03-2012)

Identifiants

Citer

Frédéric Bayart, Yanick Heurteaux. Boundary multifractal behaviour for harmonic functions in the ball. 2011. ⟨hal-00635997v1⟩
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