F. G. Avkhadiev, “A geometric description of domains whose Hardy constant is equal to 1/4”, Izv. Math., 78:5 (2014), 855

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Izvestiya: Mathematics, 2014, Volume 78, Issue 5, Pages 855–876
DOI: https://doi.org/10.1070/IM2014v078n05ABEH002710 (Mi im8121)

This article is cited in 34 scientific papers (total in 35 papers)

A geometric description of domains whose Hardy constant is equal to 1/4

F. G. Avkhadiev

Kazan (Volga Region) Federal University

English version PDF (370 kB) Citations (35) Russian version article

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DOI: https://doi.org/10.1070/IM2014v078n05ABEH002710

Abstract: We give a geometric description of families of non-convex planar and spatial domains in which the following Hardy inequality holds: the Dirichlet integral of any smooth compactly supported function

f

$f$ on the domain is greater than or equal to one quarter of the integral of

f^{2} (x) / δ^{2} (x)

$f^2(x)/\delta^2(x)$ , where

δ (x)

$\delta(x)$ is the distance from

x

$x$ to the boundary of the domain. Our geometric description is based analytically on new one-dimensional Hardy-type inequalities with special weights and on new constants related to these inequalities and hypergeometric functions.

Keywords: Hardy inequalities, non-convex domains, hypergeometric functions, torsional rigidity.

Funding agency	Grant number
Russian Foundation for Basic Research	14-01-00351-a 12-01-00636-a

Received: 16.04.2013
Revised: 10.02.2014

Bibliographic databases:

Document Type: Article

UDC: 517.5+517.518.28

MSC: 26D10, 33C20

Language: English

Original paper language: Russian

Citation: F. G. Avkhadiev, “A geometric description of domains whose Hardy constant is equal to 1/4”, Izv. Math., 78:5 (2014), 855–876

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/im8121

https://doi.org/10.1070/IM2014v078n05ABEH002710

https://www.mathnet.ru/eng/im/v78/i5/p3

This publication is cited in the following 35 articles:

R. G. Nasibullin, “Gipoteza Avkhadieva–Virtsa o nailuchshikh konstantakh Brezisa–Markusa”, Matem. sb., 216:4 (2025), 90–112
Miltiadis Paschalis, “Shape sensitivity of the Hardy constant involving the distance from a boundary submanifold”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 154:2 (2024), 408
Gerassimos Barbatis, Trends in Mathematics, 4, Modern Problems in PDEs and Applications, 2024, 3
Miltiadis Paschalis, “Hardy inequalities and uncertainty principles in the presence of a black hole”, Arch. Math., 2024
R. G. Nasibullin, “Hardy type inequalities for one weight function and their applications”, Izv. Math., 87:2 (2023), 362–388
F. G. Avkhadiev, I. R. Kayumov, S. R. Nasyrov, “Extremal problems in geometric function theory”, Russian Math. Surveys, 78:2 (2023), 211–271
F. G. Avkhadiev, “Embedding theorems related to torsional rigidity and principal frequency”, Izv. Math., 86:1 (2022), 1–31
Nasibullin R., “Hardy and Rellich Type Inequalities With Remainders”, Czech. Math. J., 72:1 (2022), 87–110
F. G. Avkhadiev, “Hardy-type inequalities with sharp constants in domains lambda-close to convex”, Siberian Math. J., 63:3 (2022), 395–411
R. G. Nasibullin, “The geometry of one-dimensional and spatial Hardy type inequalities”, Russian Math. (Iz. VUZ), 66:11 (2022), 46–78
R. G. Nasibullin, “Hardy-type inequalities for the Jacobi weight with applications”, Siberian Math. J., 63:6 (2022), 1121–1139
R. G. Nasibullin, “Sharp conformally invariant Hardy-type inequalities with remainders”, Eurasian Math. J., 12:3 (2021), 46–56
Avkhadiev F., “Selected Results and Open Problems on Hardy-Rellich and Poincare-Friedrichs Inequalities”, Anal. Math. Phys., 11:3 (2021), 134
F. G. Avkhadiev, “Properties and applications of the distance functions on open sets of the Euclidean space”, Russian Math. (Iz. VUZ), 64:4 (2020), 75–79
R. G. Nasibullin, R. V. Makarov, “Hardy's inequalities with remainders and lamb-type equations”, Siberian Math. J., 61:6 (2020), 1102–1119
Makarov V R., Nasibullin R.G., “Hardy Type Inequalities and Parametric Lamb Equation”, Indag. Math.-New Ser., 31:4 (2020), 632–649
Avkhadiev F.G., “A Strong Form of Hardy Type Inequalities on Domains of the Euclidean Space”, Lobachevskii J. Math., 41:11, SI (2020), 2120–2135
R. G. Nasibullin, “Brezis–Marcus type inequalities with Lamb constant”, Sib. elektron. matem. izv., 16 (2019), 449–464
F. G. Avkhadiev, “Conformally invariant inequalities in domains in Euclidean space”, Izv. Math., 83:5 (2019), 909–931
R. Nasibullin, “A geometrical version of Hardy-Rellich type inequalities”, Math. Slovaca, 69:4 (2019), 785–800

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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Abstract page:	1028
Russian version PDF:	215
English version PDF:	26
References:	182
First page:	137

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