On the $n$-harmonic radius of domains in the n-dimensional Euclidean space
Dalʹnevostočnyj matematičeskij žurnal, Tome 17 (2017) no. 2, pp. 246-256.

Voir la notice de l'article provenant de la source Math-Net.Ru

We extend a classical result by Lavrent’ev concerning the product of the conformal radii of planar non-overlapping domains to the case of domains in the n-dimensional Euclidean space. The conformal radius is then replaced by the n-harmonic Levitskii radius and the non-overlapping condition is replaced by a weaker geometric condition. The proofs are based on the technique of modulii of curve families. Conformal invariance of the module plays an important role in the proofs. Using the same method, we extend a classical result of Kufarev concerning the product of the conformal radii of planar non-overlapping domains in the unit disk. In addition, an inequality for n-harmonic radius of a star-shaped domain has been proved.
@article{DVMG_2017_17_2_a10,
     author = {E. G. Prilepkina},
     title = {On the  $n$-harmonic radius of domains in the n-dimensional {Euclidean} space},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {246--256},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2017_17_2_a10/}
}
TY  - JOUR
AU  - E. G. Prilepkina
TI  - On the  $n$-harmonic radius of domains in the n-dimensional Euclidean space
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2017
SP  - 246
EP  - 256
VL  - 17
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DVMG_2017_17_2_a10/
LA  - ru
ID  - DVMG_2017_17_2_a10
ER  - 
%0 Journal Article
%A E. G. Prilepkina
%T On the  $n$-harmonic radius of domains in the n-dimensional Euclidean space
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2017
%P 246-256
%V 17
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DVMG_2017_17_2_a10/
%G ru
%F DVMG_2017_17_2_a10
E. G. Prilepkina. On the  $n$-harmonic radius of domains in the n-dimensional Euclidean space. Dalʹnevostočnyj matematičeskij žurnal, Tome 17 (2017) no. 2, pp. 246-256. http://geodesic.mathdoc.fr/item/DVMG_2017_17_2_a10/

[1] V. N. Dubinin, Emkosti kondensatorov i simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo, Dalnauka, Vladivostok, 2009

[2] V. N. Dubinin, “Neravenstvo Krausa dlya mnogolistnykh funktsii”, Matem. zametki, 102:4 (2017), 559–-564 | DOI | MR

[3] B. E. Levitskii, “Privedennyi p-modul i vnutrennii p-garmonicheskii radius”, Dokl. AN SSSR., 316:4 (1991), 812–815 | MR

[4] C. Bandle, M. Flucher, “Harmonic radius and concentration of energy, hyperbolic radius and Liouvilles equations $\Delta U=0$ and $\Delta U=U^{\frac{n+2}{n-2}}$”, SIAM Review, 38:2 (1996), 191–238 | DOI | MR | Zbl

[5] W. Wang, “N-Capacity, N-harmonic radius and N-harmonic transplantation”, J. Math. Anal. Appl., 327:1 (2007), 155–174 | DOI | MR | Zbl

[6] V. N. Dubinin, E. G. Prilepkina, “Ob ekstremalnom razbienii prostranstvennykh oblastei”, Analiticheskaya teoriya chisel i teoriya funktsii. 15, Zap. nauchn. sem. POMI, 254 (1998), 95–107

[7] K. A. Gulyaeva, S. I. Kalmykov, E. G. Prilepkina, “Extremal decomposition problems in the Euclidean space”, International Journal of Mathematical Analysis, 9:56 (2015), 2763–2773 | DOI

[8] S. Kalmykov, E Prilepkina, “Extremal decomposition problems for $p$-harmonic radius”, Analysis Mathematica, 43:1 (2017), 49–65 | DOI | MR

[9] C. I. Kalmykov, E. G. Prilepkina, “O p-garmonicheskom radiuse Robena v evklidovom prostranstve”, Zap. nauchn. sem. POMI, 449 (2016), 196–213

[10] V. N. Dubinin, “Capacities and geometric transformations of subsets in n-space”, Geom. Funct. Anal., 3 (1993), 342–369 | DOI | MR | Zbl

[11] A. Yu. Solynin, “Continuous symmetrization via polarization”, Algebra i analiz, 24:1 (2012), 157–222 | MR

[12] J. Sarvas, “Symmetrization of condensers in n-space”, Ann. Acad. Sci. Fenn, Ser AI, 522 (1972), 1–44 | MR

[13] E. V. Kostyuchenko, E. G. Prilepkina, “O polyarizatsii otnositelno gipersfery”, Dalnevost. matem. zhurn., 5:1 (2004), 22–29 | MR

[14] B. Fuglede, “Extremal length and functional completion”, Acta Mathematica, 98:1 (1957), 171–219 | DOI | MR | Zbl

[15] M. Vuorinen, “Conformal geometry and quasiregular mappings”, Lecture Notes in Mathematics, Springer-Verlag, 1988 | DOI | MR

[16] V. A. Shlyk, “O ravenstve $p$-emkosti i $p$-modulya”, Sib. matem. zhurn., 34:6 (1993), 216–221 | MR | Zbl

[17] V. G. Mazya, Prostranstva S. L. Soboleva, Iz-vo Leningradskogo universiteta, Leningrad, 1985 | MR