О периодических краевых задачах для нелокального уравнения Пуассона
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Ключевые слова:
инволюция, нелокальный оператор, уравнение Пуассона, оператор Лапласа, периодическая задача, задача Дирихле, задача Неймана, собственные функции, собственные значения.Аннотация
В данной работе в единичном шаре изучаются краевые задачи с преобразованными аргументами. Преобразование аргументов задаются с помощью отображения типа инволюции. Эти отображения участвуют и в уравнении, и в краевых условиях. Рассматриваемое уравнение является нелокальным аналогом уравнения Пуассона. Краевые условия задаются в виде связи значении искомой функции в верхней полусферы со значением нижней полусферы. Эти условия обобщают известные периодические условия для шаровых областей. При исследовании краевых задач используются свойства инволютивных отображений. Рассматриваемые задачи решаются сведением их к аналогам краевых задач с периодическими условиями для классического уравнения Пуассона. Используя известные утверждения для периодических задач для рассматриваемых задач доказаны теоремы о существовании и единственности решения. Найдены точные условия разрешимости исследуемых задач. Изучены также спектральные вопросы, связанные с периодическими задачами. Найдены собственные функции и собственные значения этих задач.
Библиографические ссылки
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Sadybekov M.A., Turmetov B.Kh. On analogues of periodic boundary value problems for the Laplace operator in a ball// Eurasian Mathematical Journal. – 2012. – Vol.3, No.1. – P.143 – 146.
Sadybekov M.A., Turmetov B.Kh. On an analog of periodic boundary value problems for the Poisson equation in the disk // Differential Equations. – 2014. – Vol. 50, No. 2. – P. 268 – 273. https://doi.org/10.1134/S0012266114020153.
Karachik V.V., Turmetov B.Kh. Solvability of one nonlocal Dirichlet problem for the Poisson equation// Novi sad journal of mathematics.– 2020.– Vol. 50, No. 1. – P.67 - 88. http://doi.org/10.30755/NSJOM.08942.
Sadybekov M.A., Dukenbayeva A.A. On a nonlocal boundary value problem for the Laplace operator, which is a multidimensional generalisation of the Samarskii-Ionkin problem // News of the Khoja Akhmet Yassawi KazakhTurkish international university. Mathematics, physics, computer science. – 2018. – Vol. 1, №1(4). – P. 81–83.
Sadybekov M.A., Dukenbayeva A.A. Direct and inverse problems for the Poisson equation with equality of flows on a part of the boundary // Complex Variables and Elliptic Equations. – 2019. – Vol. 64, №5. – P. 777-791. https://doi.org/10.1080/17476933.2018.1517340
Sadybekov M.A., Dukenbayeva A.A. On boundary value problem of the Samarskii-Ionkin type for the Laplace operator in a ball // Kazakh Mathematical Journal. – 2020. – Vol. 20, №1. – P. 84–94.
Sadybekov M.A., Dukenbayeva A.A. On boundary value problem of the Samarskii-Ionkin type for the Laplace operator in a ball // Complex Variables and Elliptic Equations. – 2022.– Vol. 67, № 2 – P. 369–383. https://doi.org/10.1080/17476933.2020.1828377
Sadybekov M.A., Turmetov B.Kh., Torebek B.T. Solvability of nonlocal boundary-value problems for the Laplace equation in the ball // Electronic Journal of Differential Equations. – 2014. – Vol. 2014, No. 157. – P. 1–14.
Sadybekov M.A., Yessirkegenov N.A. On a generalised SamarskiiIonkin type problem for the Poisson equation // Kazakh Mathematical Journal. – 2017. – Vol. 17, No. 1. – P. 115–116.
Turmetov B.Kh., Koshanova M., Usmanov K. About solvability of some boundary value problems for Poisson equation in the ball conditions // Filomat. – 2018. – Vol. 32, No. 3. – P. 939-946.https://doi:10.2298/FIL1803939K
Turmetov B.Kh. Generalization of the Robin Problem for the Laplace Equation // Differential Equations.– 2019. – Vol. 55, No. 9. – P. 1134–1142. https://doi.org/10.1134/S0012266119090027
Yessirkegenov N. Spectral properties of the generalized Samarskii Ionkin type problems // Filomat. – 2018. – Vol. 32, No. 3. – P. 1019–1024. https://doi:10.2298/FIL1803019Y
Kal’menov T.S., Iskakova U.A. A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation // Dokl Math. – 2007. – Vol. 75, No. 3. – P. 370-373. https://doi.org/10.1134/S1064562407030118
Kal’menov T.S., Iskakova U.A. A method for solving the Cauchy problem for the Laplace equation // Dokl Math. – 2008. – Vol. 78, No. 3. – P. 874-876. https:// doi: 10.1134/S1064562408060185
Yarka U., Fedushko S., Vesely P. The Dirichlet Problem for the Perturbed Elliptic Equation//Mathematics. – 2020. – Vol.8, No.2108. – P.1–13. https://doi.org/10.3390/math8122108
Karachik V., Sarsenbi A., Turmetov B. On the solvability of the main boundary value problems for a nonlocal Poisson equation// Turkish Journal of Mathematics. – 2019. – Vol. 43, No. 3. – P. 1604 – 1625. https:// doi: 10.3906/mat-1901-71
Turmetov B., Karachik V. On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution//Symmetry. – 2021. – Vol.13, No.1781.– P. 1 – 20. https:// doi.org/10.3390/sym13101781
Turmetov B. Kh., Karachik V. V. On the solvability of Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution// Vestn. Udmurt University. Mat. Fur. Computer. Sciences. – 2021. – Т. 31, No. 4. – C. 651 – 667.https://doi.org/10.35634/vm210409
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Sadybekov M.A., Torebek B.T., Turmetov B.Kh.. Representation of Green’s function of the Neumann problem for a multi-dimensional ball // Complex Variables and Elliptic Equations. – 2016. – Vol. 61, № 1. – P.104–123. https://doi.org/10.1080/17476933.2015.1064402
