Some estimates for elliptic systems generalizing the Bitsadze system of equations
This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space o...
| Автори: | , |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Kazan Federal University
2024-04-01
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| Серія: | Учёные записки Казанского университета. Серия Физико-математические науки |
| Предмети: | |
| Онлайн доступ: | https://uzakufismat.elpub.ru/jour/article/view/36 |
| _version_ | 1827072645475074048 |
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| author | S. Baizaev R. N. Barotov |
| author_facet | S. Baizaev R. N. Barotov |
| author_sort | S. Baizaev |
| collection | DOAJ |
| description | This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like. |
| first_indexed | 2024-04-24T08:50:56Z |
| format | Article |
| id | doaj.art-5cda373020594f85b992a0bb2e5dddb1 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2025-03-20T00:57:36Z |
| publishDate | 2024-04-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета. Серия Физико-математические науки |
| spelling | doaj.art-5cda373020594f85b992a0bb2e5dddb12024-10-07T15:29:53ZengKazan Federal UniversityУчёные записки Казанского университета. Серия Физико-математические науки2541-77462500-21982024-04-011661223510.26907/2541-7746.2024.1.22-3531Some estimates for elliptic systems generalizing the Bitsadze system of equationsS. Baizaev0R. N. Barotov1Tajik State University of Law, Business and PoliticsKhujand State University named after Academician B. GafurovThis article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.https://uzakufismat.elpub.ru/jour/article/view/36elliptic systemfunctions bounded and uniformly h¨older continuousa priori estimatenoetherian property |
| spellingShingle | S. Baizaev R. N. Barotov Some estimates for elliptic systems generalizing the Bitsadze system of equations Учёные записки Казанского университета. Серия Физико-математические науки elliptic system functions bounded and uniformly h¨older continuous a priori estimate noetherian property |
| title | Some estimates for elliptic systems generalizing the Bitsadze system of equations |
| title_full | Some estimates for elliptic systems generalizing the Bitsadze system of equations |
| title_fullStr | Some estimates for elliptic systems generalizing the Bitsadze system of equations |
| title_full_unstemmed | Some estimates for elliptic systems generalizing the Bitsadze system of equations |
| title_short | Some estimates for elliptic systems generalizing the Bitsadze system of equations |
| title_sort | some estimates for elliptic systems generalizing the bitsadze system of equations |
| topic | elliptic system functions bounded and uniformly h¨older continuous a priori estimate noetherian property |
| url | https://uzakufismat.elpub.ru/jour/article/view/36 |
| work_keys_str_mv | AT sbaizaev someestimatesforellipticsystemsgeneralizingthebitsadzesystemofequations AT rnbarotov someestimatesforellipticsystemsgeneralizingthebitsadzesystemofequations |