Geodesic
On extremal domains for integral inequalities in the Euclidean space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2019), pp. 80-84 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On domains of the Euclidean space we consider Hardy and Rellich type inequalities with weight functions depending on the distance to the boundary of the domain. We show that extremal domains are not single for some known inequalities with sharp estimates of constants. We describe a family of extremal domains for Hardy type inequalities of a general form. On plane domains we study a new Rellich type inequality having a similar property.
Keywords: Hardy inequality, Rellich inequality. 
@article{IVM_2019_6_a7,
     author = {F. G. Avkhadiev},
     title = {On extremal domains for integral inequalities in the {Euclidean} space},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {80--84},
     year = {2019},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2019_6_a7/}
}
TY  - JOUR
AU  - F. G. Avkhadiev
TI  - On extremal domains for integral inequalities in the Euclidean space
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2019
SP  - 80
EP  - 84
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/IVM_2019_6_a7/
LA  - ru
ID  - IVM_2019_6_a7
ER  - 
%0 Journal Article
%A F. G. Avkhadiev
%T On extremal domains for integral inequalities in the Euclidean space
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2019
%P 80-84
%N 6
%U http://geodesic.mathdoc.fr/item/IVM_2019_6_a7/
%G ru
%F IVM_2019_6_a7
F. G. Avkhadiev. On extremal domains for integral inequalities in the Euclidean space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2019), pp. 80-84. http://geodesic.mathdoc.fr/item/IVM_2019_6_a7/

[1] Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge Univ. Press, Cambridge, 1934 | MR

[2] Rellich F., Perturbation theory of eigenvalue problems, Gordon and Breach, New York–London–Paris, 1969 | MR | Zbl

[3] Caldiroli P., Musina R., “Rellich inequalities with weights”, Calc. Var., 45 (2012), 147–164 | DOI | MR | Zbl

[4] Balinsky A. A., Evans W. D., Lewis R. T., The analysis and geometry of Hardy's inequality, Universitext, Springer, Heidelberg–New York–Dordrecht–London, 2015 | DOI | MR | Zbl

[5] Pommerenke Ch., “Uniformly perfect sets and the Poincaré metric”, Arch. Math., 32 (1979), 192–199 | DOI | MR | Zbl

[6] Avkhadiev F. G., “Neravenstva tipa Khardi v ploskikh i prostranstvennykh otkrytykh mnozhestvakh”, Tr. matem. in-t. im. V. A. Steklova, 255, 2006, 8–18 | Zbl

[7] Avkhadiev F. G., Wirths K.-J., “Weighted Hardy inequalities with sharp constants”, Lobachevskii J. Math., 31:1 (2010), 1–7 | DOI | MR | Zbl

[8] Avkhadiev F. G., Wirths K.-J., “Sharp Hardy-type inequalities with Lamb's constants”, Bull. Belg. Math. Soc. Simon Stevin, 18 (2011), 723–736 | MR | Zbl

[9] Avkhadiev F. G., “Rellich type inequalities with weights in plane domains”, Lobachevskii J. Math., 39:4 (2018), 1–8 | MR

[10] Avkhadiev F. G., “O neravenstvakh Rellikha v evklidovom prostranstve”, Izv. vuzov. Matem., 2018, no. 8, 83–87

[11] Avkhadiev F. G., “Hardy-Rellich inequalities in domains of the Euclidean space”, J. Math. Anal. Appl., 442 (2016), 469–484 | DOI | MR | Zbl