On the Neumann-Poincaré operator

Král, Josef, and Medková, Dagmar. "On the Neumann-Poincaré operator." Czechoslovak Mathematical Journal 48.4 (1998): 653-668. <http://eudml.org/doc/30444>.

@article{Král1998,
abstract = {Let $\Gamma $ be a rectifiable Jordan curve in the finite complex plane $\mathbb \{C\}$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma )$ the space of all complex-valued functions on $\Gamma $ which are square integrable w.r. to the arc-length on $\Gamma $. Let $L^2(\Gamma )$ stand for the space of all real-valued functions in $L^2_C (\Gamma )$ and put \[ L^2\_0 (\Gamma ) = \lbrace h \in L^2 (\Gamma )\; \int \_\{\Gamma \} h(\zeta ) |\mathrm \{d\}\zeta | =0\rbrace . \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma )$, the Neumann-Poincaré operator $C_1^\{\Gamma \}$ sending each $h \in L^2 (\Gamma )$ into \[ C\_1^\{\Gamma \} h(\zeta \_0) := \Re (\pi \mathrm \{i\})^\{-1\} \mathop \{\mathrm \{P\}. V.\}\int \_\{\Gamma \} \frac\{h(\zeta )\}\{\zeta -\zeta \_0\} \mathrm \{d\}\zeta , \quad \zeta \_0 \in \Gamma , \] is bounded on $L^2(\Gamma )$. We show that the inclusion \[ C\_1^\{\Gamma \} (L^2\_0 (\Gamma )) \subset L^2\_0 (\Gamma ) \] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma $.},
author = {Král, Josef, Medková, Dagmar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cauchy’s singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David; Cauchy's singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David},
language = {eng},
number = {4},
pages = {653-668},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Neumann-Poincaré operator},
url = {http://eudml.org/doc/30444},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Král, Josef
AU - Medková, Dagmar
TI - On the Neumann-Poincaré operator
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 4
SP - 653
EP - 668
AB - Let $\Gamma $ be a rectifiable Jordan curve in the finite complex plane $\mathbb {C}$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma )$ the space of all complex-valued functions on $\Gamma $ which are square integrable w.r. to the arc-length on $\Gamma $. Let $L^2(\Gamma )$ stand for the space of all real-valued functions in $L^2_C (\Gamma )$ and put \[ L^2_0 (\Gamma ) = \lbrace h \in L^2 (\Gamma )\; \int _{\Gamma } h(\zeta ) |\mathrm {d}\zeta | =0\rbrace . \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma )$, the Neumann-Poincaré operator $C_1^{\Gamma }$ sending each $h \in L^2 (\Gamma )$ into \[ C_1^{\Gamma } h(\zeta _0) := \Re (\pi \mathrm {i})^{-1} \mathop {\mathrm {P}. V.}\int _{\Gamma } \frac{h(\zeta )}{\zeta -\zeta _0} \mathrm {d}\zeta , \quad \zeta _0 \in \Gamma , \] is bounded on $L^2(\Gamma )$. We show that the inclusion \[ C_1^{\Gamma } (L^2_0 (\Gamma )) \subset L^2_0 (\Gamma ) \] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma $.
LA - eng
KW - Cauchy’s singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David; Cauchy's singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David
UR - http://eudml.org/doc/30444
ER -