Математическая физика и компьютерное моделирование2587-63252587-690210.15688/mpcm.jvolsu.2024.3.1OtherГАРМОНИЧЕСКИЕ ПОТЕНЦИАЛЫ НА НЕКОМПАКТНЫХ РИМАНОВЫХ МНОГООБРАЗИЯХHARMONIC POTENTIALS ON NON-COMPACT RIEMANNIAN MANIFOLDSЛосевАлександр ГеоргиевичLosevAlexanderalexander.losev@volsu.ru0000-0002-1072-8375Volgograd State University (Volgograd, Russian Federation)Волгоградский государственный университет (Волгоград, Российская Федерация)050120242736140409202423092024https://creativecommons.org/licenses/by/4.0/CC BY 4.0

Работа выполнена в рамках тематики, посвященной асимптотическому поведению решений дифференциальных уравнений в частных производных на некомпактных римановых многообразиях. Наиболее популярными разделами подобных исследований являются теоремы типа Лиувилля о тривиальности пространств решений эллиптических уравнений на некомпактных римановых многообразиях, а также вопросы разрешимости краевых задач. Считающаяся в настоящее время классической формулировка теоремы Лиувилля утверждает, что всякая ограниченная гармоническая функция в 𝑅𝑛 есть тождественная постоянная. В последнее время наметилась тенденция к более общему подходу к теоремам типа Лиувилля, а именно, оцениваются размерности различных пространств решений линейных уравнений эллиптического типа. В частности, в работе А.А. Григорьяна (1990) была доказана точная оценка размерностей пространств ограниченных гармонических функций на некомпактных римановых многообразиях в терминах массивных множеств. Вообще, исследования последних десятилетий показали крайне высокую эффективность применения емкостной техники в решении указанных задач. Данная работа посвящена развитию емкостной техники, связанной с понятием массивного множества, в исследовании асимптотического поведения гармонических функций на некомпактных римановых многообразиях.

The work is carried out within the framework of the topic devoted to the asymptotic behavior of solutions of partial differential equations on non-compact Riemannian manifolds. The most popular sections of such studies are theorems of the Liouville type on the triviality of spaces of solutions of elliptic equations on non-compact Riemannian manifolds, as well as questions of solvability of boundary value problems. The currently considered classical formulation of the Liouville theorem states that any bounded harmonic function in 𝑅𝑛 is identically constant. Recently, a tendency has emerged towards a more general approach to theorems of the Liouville type, namely, the dimensions of various spaces of solutions of linear equations of elliptic type are estimated. In particular, in the work by A.A. Grigory’an (1990), an exact estimate of the dimensions of spaces of bounded harmonic functions on non-compact Riemannian manifolds in terms of massive sets was proved. In general, studies of the last decades have shown the extremely high efficiency of using capacitive techniques in solving the above problems. This work is devoted to the development of capacitive techniques related to the concepts of a massive set in the study of the asymptotic behavior of harmonic functions on non-compact Riemannian manifolds.

гармонические функциитеоремы типа Лиувиллянекомпактные римановы многообразиямассивные множестватеория потенциалаharmonic functionsLiouville type theoremsnoncompact Riemannian manifoldsmassive setspotential theoryGrigoryan A.A. O liuvillevykh teoremakh dlya garmonicheskikh funktsiy s konechnym integralom Dirikhle [On Liouville Theorems for Harmonic Functions with Finite Dirichlet Integral]. Mat. sb., 1987, vol. 132, no. 4, pp. 496-516.Grigoryan A.A. O razmernosti prostranstv garmonicheskikh funktsiy [Dimension of Spaces of Harmonic Functions]. Mat. zametki, 1990, vol. 48, no. 5, pp. 55-60.Grigoryan A.A., Losev A.G. O razmernosti prostranstv resheniy statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh [On the Dimension of the Spaces of Solutions of the Stationary Schrodinger Equation on Non-Compact Riemannian Manifolds]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2017, vol. 20, no. 3, pp. 34-42. DOI: https://doi.org/10.15688/mpcm.jvolsu.2017.3.3Grigoryan A.A. O sushchestvovanii polozhitelnykh fundamentalnykh resheniy uravneniya Laplasa na rimanovykh mnogoobraziyakh [About Existing of Positive Fundomental Solutions of Laplass’s Equation on Riemannian Manifolds]. Mat. sb., 1985, vol. 128, no. 3, pp. 354-363.Zubankova K.A., Mazepa E.A., Poluboyarova N.M. Ob asimptoticheskom povedenii resheniy statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh [On the Asymptotic Behavior of Solutions of the Stationary Schrodinger Equation on Non-Compact Riemannian Manifolds]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2023, vol. 26, no. 4, pp. 18-30. DOI: https://doi.org/10.15688/mpcm.jvolsu.2023.4.2Kondrashov A.N. O edinstvennosti resheniy uravneniya Beltrami s zadannoy veshchestvennoy chastyu na granitse [On the Uniqueness of Solutions of the Beltrami Equation with a Given Real Part on the Boundary]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2024, vol. 27, no. 1, pp. 5-16. DOI: https://doi.org/10.15688/mpcm.jvolsu.2024.1.1Korolkov S.A., Losev A.G. Resheniya ellipticheskikh uravneniy na rimanovykh mnogoobraziyakh s kontsami [Solutions of Elliptic Equations on Riemannian Manifolds with Ends]. Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1, Matematika. Fizika, 2011, no. 1, pp. 23-40.Losev A.G., Mazepa E.A. Ogranichennye resheniya uravneniya Shredingera na rimanovykh proizvedeniyakh [Bounded Solutions of the Schrodinger Equation on Riemannian Products]. Algebra i analiz, 2001, vol. 13, no. 1, pp. 84-110.Cheng S.Y., Yau S.T. Differential Equations on Riemannian Manifolds and Their Geometric Applications. Comm. Pure and Appl. Math., 1975, vol. 28, no. 3, pp. 333-354.Grigor’yan A. Analytic and Geometric Background of Recurrence and Non-Explosion of the Brownian Motion on Riemannian Manifolds. Bulletin of Amer. Math. Soc., 1999, no. 36, pp. 135-249.Grigor’yan A., Papageorgiou E., Zhang H.-W. Asymptotic Behaviour of the Heat Semigroup on Certain Riemannian Manifolds. From Classical Analysis to Analysis on Fractals. A Tribute to Robert Strichartz, 2023, vol. 1, pp. 165-179.Korolkov S.A., Losev A.G. Generalized Harmonic Functions of Riemannian Manifolds with Ends. Mathematische Zeitschrift, 2012, iss. 272, no. 1–2, pp. 459-472.Li P., Tam L.-F. Harmonic Functions and the Structure of Complete Manifolds. J. Diff. Geom., 1992, vol. 35, no. 2, pp. 359-383.Losev A.G., Filatov V.V. Dimensions of Solution Spaces of the Schrodinger Equation with Finite Dirichlet Integral on Non-Compact Riemannian Manifolds. Lobachevskii J. Math, 2019, vol. 40, pp. 1363-1370.Losev A.G., Filatov V.V. On Capacitary Characteristics of Noncompact Riemannian Manifolds. Russian Mathematics, 2021, vol. 65, no. 3, pp. 61-67.Sung C.-J., Tam L.-F., Wang J. Spaces of Harmonic Functions. J. London Math. Soc. (2), 2000, no. 3, pp. 789-806.Grigoryan A.A. O liuvillevykh teoremakh dlya garmonicheskikh funktsiy s konechnym integralom Dirikhle [On Liouville Theorems for Harmonic Functions with Finite Dirichlet Integral]. Mat. sb., 1987, vol. 132, no. 4, pp. 496-516.Grigoryan A.A. O razmernosti prostranstv garmonicheskikh funktsiy [Dimension of Spaces of Harmonic Functions]. Mat. zametki, 1990, vol. 48, no. 5, pp. 55-60.Grigoryan A.A., Losev A.G. O razmernosti prostranstv resheniy statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh [On the Dimension of the Spaces of Solutions of the Stationary Schrodinger Equation on Non-Compact Riemannian Manifolds]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2017, vol. 20, no. 3, pp. 34-42. DOI:https://doi.org/10.15688/mpcm.jvolsu.2017.3.3Grigoryan A.A. O sushchestvovanii polozhitelnykh fundamentalnykh resheniy uravneniya Laplasa na rimanovykh mnogoobraziyakh [About Existing of Positive Fundomental Solutions of Laplass's Equation on Riemannian Manifolds]. Mat. sb., 1985, vol. 128, no. 3, pp. 354-363.Zubankova K.A., Mazepa E.A., Poluboyarova N.M. Ob asimptoticheskom povedenii resheniy statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh [On the Asymptotic Behavior of Solutions of the Stationary Schrodinger Equation on Non-Compact Riemannian Manifolds]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2023, vol. 26, no. 4, pp. 18-30. DOI:https://doi.org/10.15688/mpcm.jvolsu.2023.4.2Kondrashov A.N. O edinstvennosti resheniy uravneniya Beltrami s zadannoy veshchestvennoy chastyu na granitse [On the Uniqueness of Solutions of the Beltrami Equation with a Given Real Part on the Boundary]. Matematicheskaya fizika i kompyuternoe modelirovanie [Mathematical Physics and Computer Simulation], 2024, vol. 27, no. 1, pp. 5-16. DOI:https://doi.org/10.15688/mpcm.jvolsu.2024.1.1Korolkov S.A., Losev A.G. Resheniya ellipticheskikh uravneniy na rimanovykh mnogoobraziyakh s kontsami [Solutions of Elliptic Equations on Riemannian Manifolds with Ends]. Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1, Matematika. Fizika, 2011, no. 1, pp. 23-40.Losev A.G., Mazepa E.A. Ogranichennye resheniya uravneniya Shredingera na rimanovykh proizvedeniyakh [Bounded Solutions of the Schrodinger Equation on Riemannian Products]. Algebra i analiz, 2001, vol. 13, no. 1, pp. 84-110.Cheng S.Y., Yau S.T. Differential Equations on Riemannian Manifolds and Their Geometric Applications. Comm. Pure and Appl. Math., 1975, vol. 28, no. 3, pp. 333-354.Grigor'yan A. Analytic and Geometric Background of Recurrence and Non-Explosion of the Brownian Motion on Riemannian Manifolds. Bulletin of Amer. Math. Soc., 1999, no. 36, pp. 135-249.Grigor'yan A., Papageorgiou E., Zhang H.-W. Asymptotic Behaviour of the Heat Semigroup on Certain Riemannian Manifolds. From Classical Analysis to Analysis on Fractals. A Tribute to Robert Strichartz, 2023, vol. 1, pp. 165-179.Korolkov S.A., Losev A.G. Generalized Harmonic Functions of Riemannian Manifolds with Ends. Mathematische Zeitschrift, 2012, iss. 272, no. 1–2, pp. 459-472.Li P., Tam L.-F. Harmonic Functions and the Structure of Complete Manifolds. J. Diff. Geom., 1992, vol. 35, no. 2, pp. 359-383.Losev A.G., Filatov V.V. Dimensions of Solution Spaces of the Schrodinger Equation with Finite Dirichlet Integral on Non-Compact Riemannian Manifolds. Lobachevskii J. Math, 2019, vol. 40, pp. 1363-1370.Losev A.G., Filatov V.V. On Capacitary Characteristics of Noncompact Riemannian Manifolds. Russian Mathematics, 2021, vol. 65, no. 3, pp. 61-67.Sung C.-J., Tam L.-F., Wang J. Spaces of Harmonic Functions. J. London Math. Soc. (2), 2000, no. 3, pp. 789-806.