Approximation of harmonic functions

Dahlberg, Björn E. J.

Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 97-107.

Soit $D$ un domaine borné à frontière régulière, et $u$ une fonction harmonique dans $D$. On montre que si les valeurs de $u$ à la frontière appartiennent à $L^p\left(\sigma \right)$ avec $p\ge 2$ ($\sigma$ étant la mesure de surface à la frontière), $u$ est approchable uniformément par des fonctions à variation bornée, et on montre que le résultat ne s’étend pas au cas $p<2$.

Let $u$ be harmonic in a bounded domain $D$ with smooth boundary. We prove that if the boundary values of $u$ belong to $L^p\left(\sigma \right)$, where $p\ge 2$ and $\sigma$ denotes the surface measure of $\partial D$, then it is possible to approximate $u$ uniformly by function of bounded variation. An example is given that shows that this result does not extend to $p<2$.

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