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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 5, Pages 3–26 (Mi izv8121)  

This article is cited in 20 scientific papers (total in 21 papers)

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

F. G. Avkhadiev

Kazan (Volga Region) Federal University

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 $f$ on the domain is greater than or equal to one quarter of the integral of $f^2(x)/\delta^2(x)$, where $\delta(x)$ is the distance from $x$ 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

DOI: https://doi.org/10.4213/im8121

Full text: PDF file (610 kB)
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English version:
Izvestiya: Mathematics, 2014, 78:5, 855–876

Bibliographic databases:

UDC: 517.5+517.518.28
MSC: 26D10, 33C20
Received: 16.04.2013
Revised: 10.02.2014

Citation: F. G. Avkhadiev, “A geometric description of domains whose Hardy constant is equal to 1/4”, Izv. RAN. Ser. Mat., 78:5 (2014), 3–26; Izv. Math., 78:5 (2014), 855–876

Citation in format AMSBIB
\by F.~G.~Avkhadiev
\paper A geometric description of domains whose Hardy constant is equal to~1/4
\jour Izv. RAN. Ser. Mat.
\yr 2014
\vol 78
\issue 5
\pages 3--26
\jour Izv. Math.
\yr 2014
\vol 78
\issue 5
\pages 855--876

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. F. G. Avkhadiev, “Hardy type $L_p$-inequalities in $r$-close-to-convex domains”, Russian Math. (Iz. VUZ), 59:1 (2015), 71–74  mathnet  crossref
    2. F. G. Avkhadiev, “Sharp constants in Hardy type inequalities”, Russian Math. (Iz. VUZ), 59:10 (2015), 53–56  mathnet  crossref
    3. F. G. Avkhadiev, “Integral inequalities in domains of hyperbolic type and their applications”, Sb. Math., 206:12 (2015), 1657–1681  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. F. G. Avkhadiev, “Rellich type inequalities in domains of the Euclidean space”, Russian Math. (Iz. VUZ), 60:1 (2016), 60–63  mathnet  crossref  isi
    5. F. G. Avkhadiev, “Hardy–Rellich inequalities in domains of the Euclidean space”, J. Math. Anal. Appl., 442:2 (2016), 469–484  crossref  mathscinet  zmath  isi  elib  scopus
    6. F. G. Avkhadiev, “Estimates of Hardy–Rellich constants for polyharmonic operators and their generalizations”, Ufa Math. J., 9:3 (2017), 8–17  mathnet  crossref  isi  elib
    7. R. Nasibullin, “Hardy type inequalities for fractional integrals and derivatives of Riemann-Liouville”, Lobachevskii J. Math., 38:4, SI (2017), 709–718  crossref  mathscinet  zmath  isi  scopus
    8. F. G. Avkhadiev, “The generalized Davies problem for polyharmonic operators”, Siberian Math. J., 58:6 (2017), 932–942  mathnet  crossref  crossref  isi  elib
    9. L. A. Aksent'ev, A. I. Aptekarev, A. M. Bikchentaev, V. V. Goryainov, V. N. Dubinin, A. M. Elizarov, I. R. Kayumov, A. Laptev, S. R. Nasyrov, D. V. Prokhorov, A. G. Sergeev, V. D. Stepanov, “Farit Gabidinovich Avkhadiev (on his 70th birthday)”, Russian Math. Surveys, 73:1 (2018), 181–185  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. F. G. Avkhadiev, “Sharp Hardy constants for annuli”, J. Math. Anal. Appl., 466:1 (2018), 936–951  crossref  mathscinet  zmath  isi  scopus
    11. F. G. Avkhadiev, “Integral inequalities of Hardy and Rellich in domains satisfying an exterior sphere condition”, St. Petersburg Math. J., 30:2 (2019), 161–179  mathnet  crossref  mathscinet  isi  elib
    12. F. G. Avkhadiev, “Rellich inequalities for polyharmonic operators in plane domains”, Sb. Math., 209:3 (2018), 292–319  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. R. G. Nasibullin, “Brezis–Marcus type inequalities with Lamb constant”, Sib. elektron. matem. izv., 16 (2019), 449–464  mathnet  crossref
    14. F. G. Avkhadiev, “Conformally invariant inequalities in domains in Euclidean space”, Izv. Math., 83:5 (2019), 909–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. R. Nasibullin, “A geometrical version of Hardy-Rellich type inequalities”, Math. Slovaca, 69:4 (2019), 785–800  crossref  isi
    16. F. G. Avkhadiev, R. V. Makarov, “Hardy type inequalities on domains with convex complement and uncertainty principle of Heisenberg”, Lobachevskii J. Math., 40:9, SI (2019), 1250–1259  crossref  isi
    17. R. G. Nasibullin, “Multidimensional Hardy type inequalities with remainders”, Lobachevskii J. Math., 40:9, SI (2019), 1383–1396  crossref  mathscinet  isi
    18. F. G. Avkhadiev, “Svoistva i primeneniya funktsii rasstoyaniya otkrytogo podmnozhestva v evklidovom prostranstve”, Izv. vuzov. Matem., 2020, no. 4, 87–92  mathnet  crossref
    19. R. G. Nasibullin, R. V. Makarov, “Neravenstva Khardi s dopolnitelnymi slagaemymi i uravneniya tipa Lemba”, Sib. matem. zhurn., 61:6 (2020), 1377–1397  mathnet  crossref
    20. Makarov V R., Nasibullin R.G., “Hardy Type Inequalities and Parametric Lamb Equation”, Indag. Math.-New Ser., 31:4 (2020), 632–649  crossref  mathscinet  isi
    21. 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  crossref  mathscinet  isi
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