Email this article Avkhadiev–Wirths conjecture on best Brezis–Marcus constants
- Authors: Nasibullin R.G.1
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Affiliations:
- N. I. Lobachevsky Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan, Russia
- Issue: Vol 216, No 4 (2025)
- Pages: 90-112
- Section: Articles
- URL: https://bakhtiniada.ru/0368-8666/article/view/306698
- DOI: https://doi.org/10.4213/sm10120
- ID: 306698
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Abstract
We study Hardy-type inequalities with additional terms. The constant $\lambda(\Omega)$ multiplying the additional term depends on the geometry of the multidimensional domain $\Omega$ and the numerical parameters of the problem. This constant (functional) is commonly called the Brezis–Marcus constant. Avkhadiev and Wirths [1] put forward the conjecture that, over all $n$-dimensional domains with fixed inner radius $\delta_0$, the maximum best Brezis–Marcus constant is $\lambda(B_n)$, where $B_n $ is the $n$-ball of radius $\delta_0$. We improve the previously available lower estimates for $\lambda(B_n)$, for $n=2$ and $n= 4,…,10$, which takes us closer to this conjecture. Bibliography: 18 titles.
About the authors
Ramil' Gaisaevich Nasibullin
N. I. Lobachevsky Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan, Russia
Author for correspondence.
Email: NasibullinRamil@gmail.com
Candidate of physico-mathematical sciences, Associate professor
References
- F. G. Avkhadiev, K.-J. Wirths, “On the best constants for the Brezis–Marcus inequalities in balls”, J. Math. Anal. Appl., 396:2 (2012), 473–480
- H. Brezis, M. Marcus, “Hardy's inequalities revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:1-2 (1997), 217–237
- Ф. Г. Авхадиев, “Геометрическое описание областей, для которых константа Харди равна $1/4$”, Изв. РАН. Сер. матем., 78:5 (2014), 3–26
- A. A. Balinsky, W. D. Evans, R. T. Lewis, The analysis and geometry of Hardy's inequality, Universitext, Springer, Cham, 2015, xv+263 pp.
- Р. Г. Насибуллин, “Геометрия одномерных и пространственных неравенств типа Харди”, Изв. вузов. Матем., 2022, № 11, 52–88
- S. Filippas, V. Maz'ya, A. Tertikas, “On a question of Brezis and Marcus”, Calc. Var. Partial Differential Equations, 25:4 (2006), 491–501
- M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, “A geometrical version of Hardy's inequality”, J. Funct. Anal., 189:2 (2002), 539–548
- F. G. Avkhadiev, K.-J. Wirths, “Unified Poincare and Hardy inequalities with sharp constants for convex domains”, ZAMM Z. Angew. Math. Mech., 87:8-9 (2007), 632–642
- F. G. Avkhadiev, K.-J. Wirths, “Sharp Hardy-type inequalities with Lamb's constant”, Bull. Belg. Math. Soc. Simon Stevin, 18:4 (2011), 723–736
- G. Barbatis, S. Filippas, A. Tertikas, “Refined geometric $L^p$ Hardy inequalities”, Commun. Contemp. Math., 5:6 (2003), 869–881
- C. Bandle, Isoperimetric inequalities and applications, Monogr. Stud. Math., 7, Pitman (Advanced Publishing Program), Boston, MA–London, 1980, x+228 pp.
- J. Hersch, “Sur la frequence fondamentale d'une membrane vibrante: evaluations par defaut et principe de maximum”, Z. Angew. Math. Phys., 11 (1960), 387–413
- L. Brasco, D. Mazzoleni, “On principal frequencies, volume and inradius in convex sets”, NoDEA Nonlinear Differential Equations Appl., 27:2 (2020), 12, 26 pp.
- V. Bobkov, S. Kolonitskii, “Improved Friedrichs inequality for a subhomogeneous embedding”, J. Math. Anal. Appl., 527:1 (2023), 127383, 29 pp.
- V. Bobkov, M. Tanaka, “On subhomogeneous indefinite $p$-Laplace equations in the supercritical spectral interval”, Calc. Var. Partial Differential Equations, 62:1 (2023), 22, 39 pp.
- Р. Г. Насибуллин, “Неравенства типа Харди для одной весовой функции и их применения”, Изв. РАН. Сер. матем., 87:2 (2023), 168–195
- Р. Г. Насибуллин, “Неравенства Харди для веса Якоби и их применения”, Сиб. матем. журн., 63:6 (2022), 1313–1333
- Дж. Н. Ватсон, Теория бесселевых функций, т. 1, 2, ИЛ, М., 1949, 798 с., 220 с.
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