Blaschke-Privalov theorem for polyharmonic functions.
DOI:
https://doi.org/10.31558/1817-2237.2019.1-2.12Keywords:
polyharmonicity, subharmonic and superharmonic function, Blaschke- Privalov theoremAbstract
The problem of polyharmonicity in n-dimensional space is analyzed and the generalized Blaschke-Privalov theorem for polyharmonic functions is obtained in the work. Particular attention is paid to the superharmonic function, polynomial’s order and to the corresponding mean value formulas.References
Брело М. Основи класичної теорiї потенцiалу. - М.: Мир, 1964. - 208 с.
Курант Р. Уравнения с частными производными / Р. Курант – М.: Мир, 1964. – 830 с.
Ноздрановська А.В. Теорема про середнє для функцiй спецiального виду/ А. В. Ноздрановська // Вiсник ДонНУ. – 2016. – Сер. А: Природничi науки, Т. 1. – C. 148–151.
Покровский А. В. Теоремы о среднем для решений линейных дифференциальных уравнений с частными производными / А. В. Покровский // Мат. заметки. – 1998. – Т. 64, № 2 — C. 260–272.
Привалов И. И. Субгармонические функции / И. И. Привалов – М., Л.: ОНТИ НКТП СССР, 1937. – 199 с.
Трофименко О. Д. Узагальнення теореми про середнє для полiаналiтичних функцiй у випадках кола та круга / О. Д. Трофименко // Вiсник ДонНУ. – 2009. – Сер. А: Природничi науки, Т. 1. – C. 28–31.
Хермандер Л. Анализ линейных дифференциальных операторов с частными производными/ Л. Хермандер – М.: Мир, 1986. – Т.1 – 462 с.
Nachman Aronszajn, Thomas M. Creese, Leonard J. Lipkin Polyharmonic functions. // Clarendon Press, –1983. – P. 265.
Iwasaki K. Polytopes and the mean value property / K. Iwasaki // Discrete and Comput. Geometry – 1997. – V. 17. – P. 163–189.
Iwasaki K. Invariants of finite reflection groups and the mean value problem for polytopes / K. Iwasaki // Bull. London Math. Soc. – 1999. – V. 31. – P. 477–483.
Netuka I. Mean value property and harmonic functions / I.Netuka and J.Vesely // Classical and Modern Potential Theory and Applications, Gowri Sankaran et al., ed. – Kluwer acad. Publ., 1994. – P. 359–398.
Nozdranovska A.V., Trofymenko O.D. Mean value theorem for the function of the special type/ A.V.Nozdranovska, O.D.Trofymenko. // Book of abstracts 5th international conference for young scientists on Differential equation and application dedicated to Yaroslav Lopatynsky. – 2016. – Vasy‘l Stus Donetsk National University, Vinnytsia, Т. 1. – C. 108–109.
Nozdranovska A.V. On the equivalent definition of polyharmonic functions. //International conference of young mathematicians dedicated to the 100th anniversary of Academician of Nationfl Academe of Science of Ukraine, Professor Yu. O. Mitropolskiy(1917-2008)– 2017. – Institute of Mathematics of NAS of Ukraine, Kyiv, Т. 1. – C. 36.
Pizzetti P. Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera/ P. Pizzetti // Rendiconti Lincei, serie V. – 1909. – V.18. – P. 182–185.
Ramsey T. Mean values and classes of harmonic functions / T.Ramsey and Y.Weit // Math. Proc. Camb. Dhil. Soc. – 1984. – V. 96. – P. 501–505.
Reade M. A theorem of F´edoroff / M. Reade // Duke Math. J. – 1951. – V. 18. – P. 105–109.
Reade M. On areol monogenic functions / M. Reade // Bulletin of the Amer. Math. Soc. – 1947. – V. 53. – P. 98–103.
Trofymenko O. D. Two-radii theorem for solutions of some mean value equations / O. D. Trofymenko // Мат. студiї. – 2013. – Т 40, № 2. – С. 137–143.
Trofymenko O. D. Convolution equations and mean-value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane // Journal of Mathemati cal Sciences. – 2018. – Volume 229. – Issue 1. – P.96-107.
Volchkov V. V. Integral Geometry and Convolution Equations / V. V. Volchkov. – Dordrecht/Boston/London: Kluwer Academic Publishers. 2003. – 454 p.
Volchkov V. V. Harmonic Analysis of Mean Periodic Functions on Symmetric spaces and the Heisenberg Group / V. V. Volchkov, Vit. V. Volchkov. – Series: Springer Monographs in Math., 2009. – 216 p.
Zalcman L. Mean values and differential equations / L. Zalcman // Israel J.Math. – 1973. – V. 14 – P. 339–352.
