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Title: Essential norms of a potential theoretic boundary integral operator in $L\sp 1$ (English)
Author: Král, Josef
Author: Medková, Dagmar
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 123
Issue: 4
Year: 1998
Pages: 419-436
Summary lang: English
Category: math
Summary: 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 $\sigma$. (English)
Keyword: single layer potential
Keyword: weak normal derivative
Keyword: essential norm
MSC: 31A10
MSC: 31B10
MSC: 31B20
MSC: 31B25
idZBL: Zbl 0936.31007
idMR: MR1667114
DOI: 10.21136/MB.1998.125966
Date available: 2009-09-24T21:33:55Z
Last updated: 2020-07-29
Stable URL:
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