On the Neumann-Poincaré operator

Abstract

Let Γ 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 (Г) the space of all complex-valued functions on Γ which are square integrable w.r. to the arc-length on Γ. Let L ²(Γ) stand for the space of all real-valued functions in L ² _C (Г) and put $L_0^2 (\Gamma ) = \{ h \in L^2 (\Gamma );{\text{ }}\int_\Gamma {h(\zeta )|d\zeta | = 0} \}$ Since the Cauchy singular operator is bounded on L ² _C (Г), the Neumann-Poincaré operator C ^Г₁ sending each h ∈ L ²(Γ) into $C_1^\Gamma h(\zeta {\text{o}}): = \operatorname{Re} (\pi i)^{ - 1} P.V.\int_\Gamma {\frac{{h(\zeta )}}{{\zeta - \zeta {\text{o}}}}d\zeta ,{\text{ }}\zeta {\text{o}}} \in T$, is bounded on L ²(Γ). We show that the inclusion $C_1^\Gamma (L_0^2 (\Gamma )) \subset L_0^2 (\Gamma )$ characterizes the circle in the class of all AD-regular Jordan curves Γ.