FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
On the maximum of the Moebius invariant in the four disjoint domain problem |
D. A. Kirillova |
2010, issue 1, P. 41–49 |
Abstract |
| Let $r(D,a)$ denote the conformal radius of the domain $D$ with respect to the point $a$. In this paper we obtain the supremum of the product $$ \prod_{k=1}^{4}\frac{r(D_{k},a_{k})}{|a_{k+1}-a_{k}|}, \quad a_{5}:=a_{1} $$ for all simply connected disjoint domains $D_{k}\subset\overline{\mathbb{C}}$ and points $a_{k}\in D_{k}$, $k=1,\ldots,4$. Using the method of interior variations due to $M$. Schiffer we establish the form of quadratic differential associated with extremal partition problem $\prod\limits_{k=1}^{n}r(D_{k},a_{k})|a_{k+1}-a_{k}|^{-1}\to\sup$ for arbitrary $n\ge 3$. For $n=4$ we studed the circle domains and their boundaries for the corresponding quadratic differential. |
Keywords: conformal radius, Moebius invariants, extremal partitions, quadratic differential |
Download the article (PDF-file) |
References |
| [1] G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo premennogo, Nauka, M., 1966, 628 s. [2] V. N. Dubinin, “Simmetrizaciya v geometricheskoj teorii funkcij kompleksnogo peremennogo”, Uspexi matematicheskix nauk, 49:1 (1994), 3–76. [3] G. V. Kuz'mina, “Metody geometricheskoj teorii funkcij I, II”, Algebra i analiz, 9:3 (1997), 41–103 (5, 1–50); 5, 1–50. [4] M. A. Lavrent'ev, “K teorii konformnyx otobrazhenij”, Trudy fiz. mat. in-ta im. V. A. Steklova, 5, 1934, 159–245. [5] A. K. Baxtin, G. P. Baxtina, Yu. B. Zelinskij, Topologo-algebraicheskie struktury i geometricheskie metody v kompleksnom analize, 73, In-t matematiki NAN Ukraini, 2008, 308 s. [6] G. V. Kuz'mina, “K zadache o maksimume proizvedeniya konformnyx radiusov nenalegayushhix oblastej”, Zap. nauchn. semin. LOMI, 100, 1980, 131–145. [7] S. I. Fedorov, “O maksimume odnogo konformnogo invarianta v zadache o nenalegayushhix oblastyax”, Zap. nauchn. semin. LOMI, 112, 1981, 172–183. [8] A. K. Baxtin, “Kusochno-razdelyayushhee preobrazovanie i e'kstremal'nye zadachi so svobodnymi polyusami”, Doklady RAN, 405:2 (2005), 151–153. [9] V. N. Dubinin, D. A. Kirillova, “K zadacham ob e'kstremal'nom razbienii”, Zap. nauchn. semin. POMI, 357, 2008, 54–74. [10] G. V. Kuz'mina, Moduli semejstv krivyx i kvadratichnye differencialy, Trudy Matem. in-ta im. V. A. Steklova, 139, 1980, 241 s. |