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Integral inequalities on domains of the Euclidean space for functions with non-zero traces
https://doi.org/10.26907/0021-3446-2025-5-77-83
Abstract
On plane and space domains, we present several integral inequalities for functions with non-zero boundary trace. These new inequalities for functions are generalizations of the isoperimetric inequalities from the author’s recent paper (F.G. Avkhadiev. An analog of the Poincaré metric and isoperimetric constants, Russian Mathematics, 2024, Vol. 68, No. 9, pp. 79–85).
Our theorems are formulated using hyperbolic type domains, the distance from a point to the boundary of a domain and the hyperbolic radius. We give schemes of proofs using the Poincaré metric and its properties, some hyperbolic characteristics of plane domains as well as space domains of hyperbolic type in the sense of Loewner and Nirenberg.
About the Author
F. G. Avkhadiev
Kazan Federal University
Russian Federation
Russian Federation
Farit G. Avkhadiev.
18 Kremlyovskaya str., Kazan, 420008
References
1. Авхадиев Ф.Г. Аналог метрики Пуанкаре и изопериметрические константы, Изв. вузов. Матем. (9), 92–99 (2024).
2. Rademacher H. Über partielle und totale Differenzierbarkeit I, Math. Ann. 32 (4), 340–359 (1919).
3. Balinsky A.A., Evans W.D., Lewis R.T. The analysis and geometry of Hardy’s inequality (Springer, Heidelberg - New York - Dordrecht - London, 2015).
4. Авхадиев Ф.Г. Интегральные неравенства в областях гиперболического типа и их применения, Матем. сб. 206 (12), 3–28 (2015).
5. Avkhadiev F.G. A Strong form of Hardy type inequalities on domains of the Euclidean space, Lobachevskii J. Math. 41 (11), 2120–2135 (2020).
6. Pommerenke Ch. Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (2), 192–199 (1979).
7. Avkhadiev F.G., Wirths K.-J. Schwarz-Pick type inequalities (Birkhauser Verlag, Basel-Boston-Berlin, 2009).
8. Голузин Г.М. Геометрическая теория функций комплексного переменного (Наука, М., 1966).
9. Ahlfors L.V. Conformal invariants: topics in geometric function theory (McGraw-Hill, New York, 1973).
10. Bandle C. and Flucher M. Harmonic radius and concentration of energy hyperbolic radius and Liouville’s equations ΔU = eU and ΔU = Un+2/n-2 , SIAM Rev. 38 (2), 191–238 (1996).
11. Авхадиев Ф.Г. Конформно инвариантные неравенства (Казанск. ун-т, Казань, 2020).
12. Osgood B. Some properties of f’’/f’ and the Poincaré metric, Indiana Univ. Math. J. 31 (2), 449–461 (1982).
13. Loewner C. and Nirenberg L. Partial differential equations invariant under conformal or projective transformations, Contribution to Anal., 245–272 (1974).
14. Авхадиев Ф.Г. Конформно инвариантные неравенства в областях евклидова пространства, Изв. РАН. Сер. матем. 83 (5), 3–26 (2019).
15. Avkhadiev F.G. Euclidean maximum moduli of plane domains and their applications, Complex Variables Ellipt. Equat. 64 (11), 1869–1880 (2019).
Review
For citations:
Avkhadiev F.G. Integral inequalities on domains of the Euclidean space for functions with non-zero traces. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;(5):77-83. (In Russ.) https://doi.org/10.26907/0021-3446-2025-5-77-83
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