Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-30-44
L. S. Pulkina
In this article, we consider a nonlocal problem with integral conditions for one-dimensional 4th order partial differential equation. A distinguishing feature of this problem is the presence of integral conditions of the 1st kind. Moreover, the kernels of these conditions depend on both spatial and time variables. We suggest a new approach which enables to overcome the difficulties arising from the form of nonlocal conditions and derive a priori estimates. Obtained estimates play a significant role when we prove the existence and uniqueness of the solution to the problem.
{"title":"A problem with nonlocal integral 1st kind conditions for 4th order partial differential equation","authors":"L. S. Pulkina","doi":"10.18287/2541-7525-2024-30-2-30-44","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-30-44","url":null,"abstract":"In this article, we consider a nonlocal problem with integral conditions for one-dimensional 4th order partial differential equation. A distinguishing feature of this problem is the presence of integral conditions of the 1st kind. Moreover, the kernels of these conditions depend on both spatial and time variables. We suggest a new approach which enables to overcome the difficulties arising from the form of nonlocal conditions and derive a priori estimates. Obtained estimates play a significant role when we prove the existence and uniqueness of the solution to the problem.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"9 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-81-94
K. M. Fedorov, A. Gilmanov, A. P. Shevelyov, A. A. Izotov, A. V. Kobyashev
Tracer tests of oil reservoirs have become widely used to assess the parameters of low resistance channels, leading to premature increasing of production water cut. The existing analytical methods of their interpretation do not consider the dissipation of the tracer. The article proposes a methodology of tracer tests interpretation in the presence of low resistance channels, which considers the configuration of the tracer slug in the producer. The developed method is based on solving the inverse problem of tracer filtration in the low resistance channel and for the first time considers its dissipation. The formulation of the inverse problem of tracer filtration in the low resistance channel is based on the use of the tracer transfer equation in the channel, the ratio for the reagent flow in the channel and the reservoir, Darcy’s law and the relationship of the volume of the tracer slug with its linear size. The algorithm of numerical determination of the tracer dissipation coefficient by solving the optimization problem by the gradient descent is given. The developed algorithm has been tested on the example of the tracer tests interpretation for two producers of one of the Western Siberia fields. The lengths of each channel for the selected wells are determined. It is shown that the error of comparison of calculated and field data does not exceed 7%. It was found that the length of the shortest channel corresponds to the distance between injector and producer, other channels have a longer length and can be formed later.
{"title":"Solution of the inverse problem of tracer tests interpretation results for oil reservoirs in the presence of low resistance channels","authors":"K. M. Fedorov, A. Gilmanov, A. P. Shevelyov, A. A. Izotov, A. V. Kobyashev","doi":"10.18287/2541-7525-2024-30-2-81-94","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-81-94","url":null,"abstract":"Tracer tests of oil reservoirs have become widely used to assess the parameters of low resistance channels, leading to premature increasing of production water cut. The existing analytical methods of their interpretation do not consider the dissipation of the tracer. The article proposes a methodology of tracer tests interpretation in the presence of low resistance channels, which considers the configuration of the tracer slug in the producer. The developed method is based on solving the inverse problem of tracer filtration in the low resistance channel and for the first time considers its dissipation. The formulation of the inverse problem of tracer filtration in the low resistance channel is based on the use of the tracer transfer equation in the channel, the ratio for the reagent flow in the channel and the reservoir, Darcy’s law and the relationship of the volume of the tracer slug with its linear size. The algorithm of numerical determination of the tracer dissipation coefficient by solving the optimization problem by the gradient descent is given. The developed algorithm has been tested on the example of the tracer tests interpretation for two producers of one of the Western Siberia fields. The lengths of each channel for the selected wells are determined. It is shown that the error of comparison of calculated and field data does not exceed 7%. It was found that the length of the shortest channel corresponds to the distance between injector and producer, other channels have a longer length and can be formed later.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"50 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-54-66
M. A. Fomchenkova, L. V. Stepanova
The article proposes and implements a procedure for reconstructing the asymptotic series expansion of stress, strain and displacement fields in anisotropic materials, generalizing the Williams solution for linearly elastic isotropic materials, based on a finite element solution to the problem of deforming a sample with a defect in an anisotropic orthotropic material in the approximation of a plane problem of elasticity theory. The stress field expansion coefficients near the crack tip in an anisotropic material are determined using an overdeterministic method originally proposed to reconstruct the asymptotic expansion from experimental data of a photoelastic study. In this paper, this method is extended to anisotropic materials with various types of symmetry and the novelty of the proposed approach lies in the reconstruction of the asymptotic expansion from the finite element solution for the stress tensor components in the nodes of the finite element grid, which allows us not to exclude their displacement fields components corresponding to the displacement of a body as an absolutely solid body. In the proposed approach, it is possible to use data from finite element calculations directly in the scheme of the overdeterministic method. It is shown that the coefficients of higher approximations are reliably determined by an overdeterministic method based on the stress field found from finite element analysis.
{"title":"Procedure of the overdeterminisctic method for finding the field expansion coefficients at the crack tip based on a finite element solution for the stress tensor components","authors":"M. A. Fomchenkova, L. V. Stepanova","doi":"10.18287/2541-7525-2024-30-2-54-66","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-54-66","url":null,"abstract":"The article proposes and implements a procedure for reconstructing the asymptotic series expansion of stress, strain and displacement fields in anisotropic materials, generalizing the Williams solution for linearly elastic isotropic materials, based on a finite element solution to the problem of deforming a sample with a defect in an anisotropic orthotropic material in the approximation of a plane problem of elasticity theory. The stress field expansion coefficients near the crack tip in an anisotropic material are determined using an overdeterministic method originally proposed to reconstruct the asymptotic expansion from experimental data of a photoelastic study. In this paper, this method is extended to anisotropic materials with various types of symmetry and the novelty of the proposed approach lies in the reconstruction of the asymptotic expansion from the finite element solution for the stress tensor components in the nodes of the finite element grid, which allows us not to exclude their displacement fields components corresponding to the displacement of a body as an absolutely solid body. In the proposed approach, it is possible to use data from finite element calculations directly in the scheme of the overdeterministic method. It is shown that the coefficients of higher approximations are reliably determined by an overdeterministic method based on the stress field found from finite element analysis.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"43 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-45-53
A. V. Suslov, E. E. Yaroslavkina
Studies have been carried out of the influence of temperature stresses on the frequencies of natural oscillations rectangular plates under different fastening conditions using analytical methods and computer modeling using the finite element method. It has been established that with increasing temperature the frequency of natural oscillations decreases. The presence of temperature stresses has significant influence on the change in oscillation frequency. The lowest ones undergo the greatest change frequencies. In addition, the shape of the vibrations changes with increasing temperature
{"title":"Study of the influence of temperature stresses to natural vibrations of the plates","authors":"A. V. Suslov, E. E. Yaroslavkina","doi":"10.18287/2541-7525-2024-30-2-45-53","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-45-53","url":null,"abstract":"Studies have been carried out of the influence of temperature stresses on the frequencies of natural oscillations rectangular plates under different fastening conditions using analytical methods and computer modeling using the finite element method. It has been established that with increasing temperature the frequency of natural oscillations decreases. The presence of temperature stresses has significant influence on the change in oscillation frequency. The lowest ones undergo the greatest change frequencies. In addition, the shape of the vibrations changes with increasing temperature","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"56 26","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141838028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-20-29
O. S. Kipkaeva
The article is devoted to the peculiarities of stability change of slow invariant manifolds of singularly perturbed systems of ordinary differential equations. It should be noted that the change of stability of invariant manifolds can proceed according to different scenarios. In addition to two well-known scenarios of this phenomenon, one more scenario is considered in this paper. To demonstrate the peculiarities of the stability change of slow invariant manifolds under this scenario, a number of examples are proposed. The existence theorem of an exact invariant manifold with stability change for some class of singularly perturbed systems of ordinary differential equations is obtained
{"title":"On one scenario for changing the stability of invariant manifolds of singularly perturbed systems","authors":"O. S. Kipkaeva","doi":"10.18287/2541-7525-2024-30-2-20-29","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-20-29","url":null,"abstract":"The article is devoted to the peculiarities of stability change of slow invariant manifolds of singularly perturbed systems of ordinary differential equations. It should be noted that the change of stability of invariant manifolds can proceed according to different scenarios. In addition to two well-known scenarios of this phenomenon, one more scenario is considered in this paper. To demonstrate the peculiarities of the stability change of slow invariant manifolds under this scenario, a number of examples are proposed. The existence theorem of an exact invariant manifold with stability change for some class of singularly perturbed systems of ordinary differential equations is obtained","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141838116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-12-19
E. S. Dolgova
The work is devoted to solving the problem of critical conditions for an autocatalytic combustion model, taking into account the consumption of reagent and oxidizer. By use the methods of geometric theory of singular perturbations, the analysis of the mathematical model of this process show that there are two main types of combustion modes: the slow combustion mode and the thermal explosion mode. The critical mode is intermediate between them. In the paper, the condition of the critical regime is obtained in the form of an asymptotic representation of the corresponding value of the system parameter reflecting the heat loss from the reaction phase
{"title":"Asymptotics of critical conditions in one combustion model","authors":"E. S. Dolgova","doi":"10.18287/2541-7525-2024-30-2-12-19","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-12-19","url":null,"abstract":"The work is devoted to solving the problem of critical conditions for an autocatalytic combustion model, taking into account the consumption of reagent and oxidizer. By use the methods of geometric theory of singular perturbations, the analysis of the mathematical model of this process show that there are two main types of combustion modes: the slow combustion mode and the thermal explosion mode. The critical mode is intermediate between them. In the paper, the condition of the critical regime is obtained in the form of an asymptotic representation of the corresponding value of the system parameter reflecting the heat loss from the reaction phase","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"3 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-7-11
S. A. Badonova
In a recent article by V.V. Kapustin a de Branges space, whose element is an expression containing the Riemann xi function, was constructed; the canonical system with a diagonal Hamiltonian and the generalized Fourier transform corresponding to the space were found. In this article we present a similar de Branges space with some preferred modifications and we provide formulas related to it; we also write down the Hamiltonian and the generalized Fourier transform.
卡普斯京（V.V. Kapustin）在最近的一篇文章中构建了一个德布朗日空间，其元素是包含黎曼 xi 函数的表达式；并找到了与该空间相对应的具有对角哈密顿的规范系统和广义傅里叶变换。在这篇文章中，我们提出了一个类似的德布朗杰斯空间，并做了一些优选的修改，还提供了与之相关的公式；我们还写下了哈密顿和广义傅里叶变换。
{"title":"On a de Branges space related to the Riemann zeta function","authors":"S. A. Badonova","doi":"10.18287/2541-7525-2024-30-2-7-11","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-7-11","url":null,"abstract":"In a recent article by V.V. Kapustin a de Branges space, whose element is an expression containing the Riemann xi function, was constructed; the canonical system with a diagonal Hamiltonian and the generalized Fourier transform corresponding to the space were found. In this article we present a similar de Branges space with some preferred modifications and we provide formulas related to it; we also write down the Hamiltonian and the generalized Fourier transform.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"14 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.18287/2541-7525-2024-30-2-67-80
A. R. Agaev, A. Gilmanov, A. P. Shevelyov
Currently, there is a problem of depletion of easily produce oil. In order to maintain hydrocarbon production rates, hard-to-recover reserves are being brought into development, a significant part of which are extra-heavy oil, the production of which takes a relatively small share in the global oilfield due to the complexity of the process. The methods existing at the moment do not allow extracting heavy and extra-heavy oil from reservoirs with a sufficient degree of efficiency. The use of such a method as microwave impact has not been widely used in the oilfield, because modeling is necessary to determine the optimal parameters of the impact. It is difficult with the number of problems associated with the complexity of the method. This article deals with the modeling of the process of microwave impact to improve the efficiency of the oil production process. The article is devoted to modeling the process of ultra-high-frequency wave impact on the oil reservoir, considering the physical and chemical parameters of fluids in the reservoir, such as thermal conductivity, dielectric permeability of oil and water (considering its salinity) in the reservoir. In the framework of the method using microwave impact for the first time determined the amount of shielding by the pipe material of this impact and determined the optimal parameters of the radiation source and the parameters of well pipe structures for effective impact on oil reservoirs. The aim of the work is to determine the optimal parameters of the source of microwaves to achieve cost-effective values of oil recovery factor. In this work the physical and mathematical model of microwave impact on the reservoir, based on the laws of electrodynamics and the density of volumetric heat generation in this equation is applied. The dependence of the magnitude of screening of microwave radiation by the production well pipe on its thickness and the dependence of the magnitude of screening of radiation by the production well pipe on the thickness of the perforation slot in this pipe and the dependence of the radius of penetration of electromagnetic waves into the formation on the absorption factor of electromagnetic radiation in the formation are obtained. The paper establishes the existence of a minimum radius of penetration of microwave radiation into the formation to achieve cost-effective values of oil recovery factor over 30%, which is 57 m, and also determined the absorption factor of microwave radiation in the formation, which allows to achieve the specified value of the radius of penetration of microwave radiation into the formation.
{"title":"Calculation of the shielding at microwave impact on oil reservoirs","authors":"A. R. Agaev, A. Gilmanov, A. P. Shevelyov","doi":"10.18287/2541-7525-2024-30-2-67-80","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-2-67-80","url":null,"abstract":"Currently, there is a problem of depletion of easily produce oil. In order to maintain hydrocarbon production rates, hard-to-recover reserves are being brought into development, a significant part of which are extra-heavy oil, the production of which takes a relatively small share in the global oilfield due to the complexity of the process. The methods existing at the moment do not allow extracting heavy and extra-heavy oil from reservoirs with a sufficient degree of efficiency. The use of such a method as microwave impact has not been widely used in the oilfield, because modeling is necessary to determine the optimal parameters of the impact. It is difficult with the number of problems associated with the complexity of the method. This article deals with the modeling of the process of microwave impact to improve the efficiency of the oil production process. The article is devoted to modeling the process of ultra-high-frequency wave impact on the oil reservoir, considering the physical and chemical parameters of fluids in the reservoir, such as thermal conductivity, dielectric permeability of oil and water (considering its salinity) in the reservoir. In the framework of the method using microwave impact for the first time determined the amount of shielding by the pipe material of this impact and determined the optimal parameters of the radiation source and the parameters of well pipe structures for effective impact on oil reservoirs. The aim of the work is to determine the optimal parameters of the source of microwaves to achieve cost-effective values of oil recovery factor. In this work the physical and mathematical model of microwave impact on the reservoir, based on the laws of electrodynamics and the density of volumetric heat generation in this equation is applied. The dependence of the magnitude of screening of microwave radiation by the production well pipe on its thickness and the dependence of the magnitude of screening of radiation by the production well pipe on the thickness of the perforation slot in this pipe and the dependence of the radius of penetration of electromagnetic waves into the formation on the absorption factor of electromagnetic radiation in the formation are obtained. The paper establishes the existence of a minimum radius of penetration of microwave radiation into the formation to achieve cost-effective values of oil recovery factor over 30%, which is 57 m, and also determined the absorption factor of microwave radiation in the formation, which allows to achieve the specified value of the radius of penetration of microwave radiation into the formation.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"11 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141837984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.18287/2541-7525-2024-30-1-82-95
A. R. Bagrov, E. K. Bashkirov
In this paper, we investigated the dynamics of systems of two and three identical qubits interacting resonantly with a selected mode of a thermal field of a lossless resonator. We found solutions of the quantum time-dependent Liouville equation for various three- and two-qubit entangled states of qubits. Based on these solutions, we calculated the criterion of the qubit entanglement — fidelity. The results of numerical calculations of the fidelity showed that increasing the average number of photons in a mode leads to a decrease in the maximum degree of entanglement. It is shown that the two-qubit entangled state is more stable with respect to external noise than the three-qubit entangled Greenberger — Horne — Zeilinger states (GHZ). Moreover, a genuine entangled GHZ-state is more stable to noise than a GHZ-like entangled state.
{"title":"Dynamics of entangled Greenberger — Horne — Zeilinger states in three qubits thermal Tavis — Cummings model","authors":"A. R. Bagrov, E. K. Bashkirov","doi":"10.18287/2541-7525-2024-30-1-82-95","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-1-82-95","url":null,"abstract":"In this paper, we investigated the dynamics of systems of two and three identical qubits interacting resonantly with a selected mode of a thermal field of a lossless resonator. We found solutions of the quantum time-dependent Liouville equation for various three- and two-qubit entangled states of qubits. Based on these solutions, we calculated the criterion of the qubit entanglement — fidelity. The results of numerical calculations of the fidelity showed that increasing the average number of photons in a mode leads to a decrease in the maximum degree of entanglement. It is shown that the two-qubit entangled state is more stable with respect to external noise than the three-qubit entangled Greenberger — Horne — Zeilinger states (GHZ). Moreover, a genuine entangled GHZ-state is more stable to noise than a GHZ-like entangled state.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"9 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140658953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.18287/2541-7525-2024-30-1-23-30
S. T. Huseynov, M. J. Aliyev
We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.
{"title":"Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations","authors":"S. T. Huseynov, M. J. Aliyev","doi":"10.18287/2541-7525-2024-30-1-23-30","DOIUrl":"https://doi.org/10.18287/2541-7525-2024-30-1-23-30","url":null,"abstract":"We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. \u0000Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.","PeriodicalId":427884,"journal":{"name":"Vestnik of Samara University. Natural Science Series","volume":"65 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140665405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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