Essential norms of a potential theoretic boundary integral operator in $L\sp 1$

Král, Josef ; Medková, Dagmar

Mathematica Bohemica, Tome 123 (1998) no. 4, pp. 419-436.

Résumé

Let $G \subset\Bbb R^m$ $(m \ge2)$ be an open set with a compact boundary $B$ and let $\sigma\ge0$ be a finite measure on $B$. Consider the space $L^1(\sigma)$ of all $\sigma$-integrable functions on $B$ and, for each $f \in L^1(\sigma)$, denote by $f \sigma$ the signed measure on $B$ arising by multiplying $\sigma$ by $f$ in the usual way. $\Cal N_{\sigma}f$ denotes the weak normal derivative (w.r. to $G$) of the Newtonian (in case $m >2$) or the logarithmic (in case $n=2$) potential of $f\sigma$, correspondingly. Sharp geometric estimates are obtained for the essential norms of the operator $\Cal N_{\sigma} - \alpha I$ (here $\alpha\in\Bbb R$ and $I$ stands for the identity operator on $L^1(\sigma)$) corresponding to various norms on $L^1(\sigma)$ inducing the topology of standard convergence in the mean w.r. to $\sigma$.

Détail

DOI : 10.21136/MB.1998.125966

Classification : 31A10, 31B10, 31B20, 31B25
Keywords: single layer potential; weak normal derivative; essential norm