AREA OF SEPARATE-HARMONICITY

Аннотация

The paper considers the concept of separate harmonicity of functions of several variables with respect to disjoint groups of variables. The separate harmonicity hulls for domains are studied. In particular, it is proven that a domain possessing barrier functions on a densesubset of theboundary isadomain of separateharmonicity. Keywords:Harmonic function, separately harmonic function, separate harmonicity hull, barrier, domainof separateharmonicity Аннотация: В работе рассматривается понятие сепаратно-гармоничности функций многих переменных по раздельным группам переменных. Исследуются оболочки сепаратно-гармоничности для областей. В частности, доказано, что область обладающего барьерными функциями на всюду плотном подмножестве границы, является областью сепаратно-гармоничности. Ключевые слова: Гармоническая функция,сепаратно-гармоническая функция,оболочка сепаратно-гармоничности,барьер,областьсепаратно-гармоничности. 1.Introduction It is known that for any flat domain in D there exists a function holomorphic in D and not analytically continued beyond this domain, i.e. every flat domain is a domain of holomorphy. In contrast, in the space n , n >1 , there exist domains from which any holomorphic function can be analytically continued into a wider domain. For example, a non- { } simply connected domain z n :1 1, every domain is not a domain of holomorphy, i.e. the class holomorphic functions of several complex variables have the effect offorced analytic continuation (see [1,p.153]). It turns out that the class of separately harmonic functions also has the property of forced analyticcontinuation. In this paper, westudythedomains ofseparately harmonicity. 2.Separateharmonicfunctions Let x Rn , y Rm , Rn Rm D G — be a domain, D E and G F — somesubsets. Supposethat, following thefunction u(x,y),initially defined on theset E F , has theproperties: a)for any fixed x0 E ,the function u(x0,y) continues harmonically inG ; b)forany fixed y 0 F thefunction u(x,y0) continues harmonically in D . 100 INTERNATIONAL MULTIDISCIPLINARY JOURNAL FOR RESEARCH & DEVELOPMENT SJIF2019:5.2222020:5.5522021:5.6372022:5.4792023:6.5632024:7,805 eISSN:2394-6334 https://www.ijmrd.in/index.php/imjrd Volume12,issue07(2025) In this case, the specified extensions and u(x,y) define a certain function on the set X =( E G )U( D F ) ,which is called a separateharmonic function on X . In the case when in E =D and F =G ,the function u(x,y) is called separately harmonic in theregion X =D G ,i.e. harmonic withrespect tothegroups ofvariables separately. The set X is not domain in general. For an arbitrary domain that cannot be represented as a product of two domains, the separately harmonic function is defined as follows: if the function u(x,y) is defined in the domain W Rn(x) Rm(y) , n,m >1, and has the following properties. { } 1) for any x 0 : x =x 0 IW , function u(x 0,y) harmonic with respect to variable y on { } thesection x =x 0 IW; { } 2) for any y 0 : y =y 0 IW , the function u(x,y 0) is harmonic in the variable x on the { } section y =y 0 IW,then itis called a separately harmonic function in region W. Theabove defined class ofseparately harmonicfunctions is denoted by h (W) (h (X)). nm nm The well-known Hartogs theorem (see [1]) states that if a function f (z,w) is holomorphic in the domain D n for fixed w and it is holomorphic in G m holomorphic in the domain D G n(z) m(w) withrespect tothe set ofvariables. In 1961, P. Lelong [2] proved the following analogue of this theorem for separately harmonic functions: if the function u(x,y) is separately harmonic in the domain D G Rn(x) Rm(y) , then the function u(x,y) is harmonic in D G with respect to theset ofvariables. Let us now consider the following general problem: let in E D Rn , F G Rm and u(x,y) separately harmonic in the set it is required to describe the X =( E G )U( D F ) theharmonicdomain functions u(x,y) be. Thisproblem was studied in theworks [3],[4]), [6],[8],[9]. In 1982, A. Zeryakhi [4] obtained the following result: let the domain of satisfying D G — space R 2(x) R 2(y) and E D , F G — be compact sets the conditions of H regularity in classes of harmonic polynomials. Then any separately harmonic function on theset X =( E G )U( D F ) isharmonically continued into thedomain { } X = (x,y) D G :w* (x,E,D)+w* (y,F,G)<1 . sh sh Usually, to continue harmonic functions, one first goes to holomorphic functions and then uses theprinciples ofholomorphic continuations. The following result plays a fundamental role in the study of continuation of harmonic functions. Preposition 1 (see [6]). Consider the space Rn(x) , embedded in Cn(z) = Rn(x)+i Rn(y) , where z =( z ,...,z ) , z =x +i y , j =1,...,n , and let 1 n j j j 101 INTERNATIONAL MULTIDISCIPLINARY JOURNAL FOR RESEARCH & DEVELOPMENT SJIF2019:5.2222020:5.5522021:5.6372022:5.4792023:6.5632024:7,805 eISSN:2394-6334 https://www.ijmrd.in/inde