| Title:
|
On the Neumann-Poincaré operator (English) |
| Author:
|
Král, Josef |
| Author:
|
Medková, Dagmar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
1998 |
| Pages:
|
653-668 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $. (English) |
| Keyword:
|
Cauchy’s singular operator |
| Keyword:
|
the Neumann-Poincaré operator |
| Keyword:
|
curves regular in the sense of Ahlfors and David |
| MSC:
|
30E20 |
| MSC:
|
47B38 |
| idZBL:
|
Zbl 0956.30018 |
| idMR:
|
MR1658229 |
| . |
| Date available:
|
2009-09-24T10:16:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127444 |
| . |
| Reference:
|
[1] M. G. Arsove: Continuous potentials and linear mass distributions.SIAM Rev. 2 (1960), 177–184. Zbl 0094.08005, MR 0143926, 10.1137/1002039 |
| Reference:
|
[2] G. David: Opérateurs intégraux singuliers sur certaines courbes du plane complexe.Ann. Scient. Éc. Nor. Sup. 17 (1984), 157–189. MR 0744071, 10.24033/asens.1469 |
| Reference:
|
[3] P.L. Duren: Theory of $H^p$ spaces.Academic Press, 1970. MR 0268655 |
| Reference:
|
[4] D. Gaier: Integralgleichungen erster Art and konforme Abbildung.Math. Z. 147 (1976), 113–129. MR 0396926, 10.1007/BF01164277 |
| Reference:
|
[5] J. Král, I. Netuka, J. Veselý: Teorie potenciálu II.Státní ped. nakl., Praha, 1972. |
| Reference:
|
[6] J. G. Krzy.z: Some remarks concerning the Cauchy operator on AD-regular curves.Annales Un. Mariae Curie-Skłodowska XLII, 7 (1988), 53–58. Zbl 0716.30034, MR 1074844 |
| Reference:
|
[7] J. G. Krzy.z: Generalized Neumann–Poincaré operator and chord-arc curves.Annales Un. Mariae Curie-Skłodowska XLIII, 7 (1989), 69–78. Zbl 0736.30028, MR 1158099 |
| Reference:
|
[8] J. G. Krzy.z: Chord-arc curves and generalized Neumann-Poincaré operator $C_1^{\Gamma }$.“Linear and Complex Analysis Problem Book 3”, Lecture Notes in Math. 1579, V. P. Havin and N. K. Nikolski (eds.), 1994, p. 418. |
| Reference:
|
[9] E. Martensen: Eine Integralgleichung für die logarithmische Gleichgewichtsbelegung und die Krümmung der Randkurve eines ebenen Gebiets.Z. angew. Math.-Mech. 72 (6) (1992), T596–T599. MR 1178329 |
| Reference:
|
[10] Ch. Pommerenke: Boundary behaviour of conformal maps.Springer-Verlag, 1992. Zbl 0762.30001, MR 1217706 |
| Reference:
|
[11] I. I. Priwalow: Randeigenschaften analytischer Funktionen.Translated from Russian, Deutscher Verlag der Wissenschaften, Berlin, 1956. Zbl 0073.06501, MR 0083565 |
| Reference:
|
[12] S. Saks: Theory of the integral.Dover Publications, New York, 1964. MR 0167578 |
| . |
