Inequalities for surface integrals of non-negative subharmonic functions

Aldred, M. P. ; Armitage, D. H.

Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 1, pp. 101-113.

Résumé

Let ${\Cal H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in $\Bbb R^N$. Let $S$ be a sphere contained in $\overline{\Omega}$, and let $\sigma$ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega$ which guarantees that there exists a constant $K$, depending only on $\Omega$ and $S$, such that $\int_Su\,d\sigma \le K\int_{\partial\Omega}u\,d\sigma$ for every $u\in {\Cal H}\cap C(\overline{\Omega})$. If this inequality holds for every such $u$, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for $K$ are given. In particular the classical value $K=2$ for convex domains is slightly improved.

Détail

Classification : 31B05
Keywords: subharmonic; surface integral