Trudy Instituta prikladnoj matematiki i mehaniki最新文献

{"title":"Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions","authors":"Ihor Skrypnik, Maria Savchenko, Yevgeniia Yevgenieva","doi":"10.37069/1683-4720-2023-37-5","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-5","url":null,"abstract":"The study of the regularity of solutions to the elliptic equations with non-standard growth has been initiated by Zhikov, Marcellini, and Lieberman, and in the last thirty years, the qualitative theory of second-order elliptic and parabolic equations has been actively developed. Equations of this type and systems of such equations arise in various problems of mathematical physics and engineering (e.g. in describing electrorheological fluids, or in image recognition and data denoising). There are two cases of the type of growth. The simple so-called ''logarithmic'' case is studied very well and there are a lot of classical results in this regard. But the so-called ''non-logarithmic'' growth differs substantially from the logarithmic case. The non-logarithmic condition introduced by Zhikov turned out to be a precise condition for the smoothness of finite functions in the corresponding Sobolev space, which makes it extremely interesting to study. But to our knowledge, there are only a few results in this direction. Zhikov and Pastukhova proved higher integrability of the gradient of solutions to the $p(x)$-Laplace equation under the non-logarithmic condition. Interior continuity, continuity up to the boundary, and Harnack's inequality to the $p(x)$-Laplace equation were proved by Alkhutov, Krasheninnikova, and Surnachev. These results were generalized by Skrypnik and Voitovich. The qualitative properties of bounded solutions of $p(x)$-Laplace equation under the non-logarithmic condition were established by Skrypnik and Yevgenieva. As for unbounded solutions, there are just a few results. Ok has proved the boundedness of minimizers of elliptic functionals of the double-phase type under some assumptions on the growth parameters. The obtained condition gives a possibility to improve the regularity results for unbounded minimizers. The weak Harnack inequality for unbounded supersolutions of the corresponding elliptic equations with generalized Orlicz growth under the so-called logarithmic conditions was proved by Benyaiche, Harjulehto, H\"{a}st\"{o} and Karppinen. In the current paper, the weak Harnack inequality for unbounded solutions to the $p(x)$-Laplace equation has been proved under the precise non-logarithmic condition on the function $p(x)$.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503528","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}

{"title":"On the surfaces moduli theory","authors":"Volodymyr Ryazanov, Evgeny Sevost'yanov","doi":"10.37069/1683-4720-2023-37-4","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-4","url":null,"abstract":"In this article we continue to develop the theory of several moduli of families of surfaces, in particular, strings (open surfaces) of various dimensions in Euclidean spaces. Since the surfaces in question can be extremely fractal (wild), the natural basis for studying them is the so-called Hausdorff measures. As is known, these moduli are the main geometric tool in the mo-dern mapping theory and related topics in geometry, topology and the theory of partial differential equations with appropriate applications to the boundary-value problems of mathematical physics in anisotropic and inhomogeneous media. In addition, this theory can also find its further applications in many other fields, including mathematics itself (nonlinear dynamics, minimal surfaces), theoretical physics (conformal field theory, string theory), and engineering (mathematical models of the filtration of gases and fluids in underground mines of water, gas and oil seams, crystal growth and others). On the basis of the proof of Lemma~1 about the connections between moduli and the Lebesgue measures, we have proved the corresponding analogue of the Fubini theorem in the terms of the moduli that extends the known V\"ais\"al\"a theorem for families of curves to families of surfaces of arbitrary dimensions. It is necessary to note specially here that the most refined place in the proof of Lemma~1 is Proposition~1 on measurable (Borel) hulls of sets in Euclidean spaces. We also prove here the corresponding Lemma~2 and Proposition~2 on families of centered spheres. Finally, in a similar way, suitable results can be also obtained for families of several spheroids.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503529","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}

{"title":"Adomian decomposition method in the theory of weakly nonlinear boundary value problems","authors":"Sergey Chuiko, Olga Nesmelova, Mykyta Popov","doi":"10.37069/1683-4720-2023-37-6","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-6","url":null,"abstract":"The problem of solvability of nonlinear boundary value problems originates from the classical theory of periodic boundary value problems for systems of ordinary differential equations, developed in the works of A. Poincare, O.M. Lyapunov, I.G. Malkin, Yu.O. Mitropolsky, A.M. Samoilenko, O.A. Boichuk and others. In the classical works of R. Bellman, J. Hale, Y.O. Mitropolsky, A.M. Samoilenko and O.A.~Boichuk, the conditions for solvability of nonlinear boundary value problems for systems of differential equations in critical cases were obtained. To find solutions to nonlinear boundary value problems for systems of differential equations in critical cases, iterative schemes using the method of simple iterations were constructed in the monographs of A.M. Samoilenko and O.A.~Boichuk. In the works of O.A. Boichuk and S.M. Chuiko, iterative schemes based on the Newton--Kantorovich scheme with quadratic convergence were constructed to find solutions to nonlinear boundary value problems, and constructive conditions for convergence were obtained. The technique for constructing approximations to solutions of weakly nonlinear boundary value problems using the Adomian de-com-po-si-tion method investigated in this paper differs from the authors' previous results in that the boundary condition, the number of components of which, in general, does not coincide with the dimension of the solution. The results obtained can be transferred to weakly nonlinear boundary value problems with a boundary condition using nonlinear bounded vector functions. The article obtains constructive conditions for solvability and a scheme for constructing solutions to a weakly nonlinear boundary value problem for an ordinary differential equation in the critical case using the Adomian decomposition method.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503527","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}

{"title":"Identification of external harmonic force parameters","authors":"Volodymyr Shcherbak, Nadiya Zhogoleva","doi":"10.37069/1683-4720-2023-37-2","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-2","url":null,"abstract":"The problem of determining the external force, which is given by the harmonic function of time and acts on a self-oscillating system of general type (Lienard oscillator) is considered. A general method of asymptotic estimation of oscillator velocity and force unknown parameters is proposed. Such problems of estimating the frequency, amplitude, and phase of an external force acting on a mechanical system are reflected in a sufficient number of publications both in past and present times. The reason for this interest lies in the use of appropriate techniques in various theoretical and engineering disciplines, for example, for mechanical systems for converting the kinetic energy of vibrations, in the problems of vibration isolation of periodic components of noise through rotating mechanisms, to compensate for harmonic disturbances in automatic control algorithms, in adaptive filtering during signal processing, and so on. In principle, the least squares method, Fourier analysis, and Laplace Transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. Despite the relative simplicity of the problem of determining the frequency, amplitude and phase of vibrations, approaches to solving them use a rather complex apparatus of modern methods of Applied Mathematics. The aim of this paper is to extend the method of synthesis of invariant relations to the problem of determining the parameters of external influence on a mechanical system. To obtain asymptotic estimates of the coefficients of external force, the method of invariant relations developed in analytical mechanics is used. Method was intended, in particular, to search for partial solutions (dependencies between variables) in problems of dynamics of rigid body with a fixed point. Modification of this method to the problems of observation theory made it possible to synthesize additional connections between known and unknown quantities of the original system that arise during the movement of its extended dynamic model. The asymptotic convergence of estimates of unknowns to their true value is proved. The results of numerical modeling of the asymptotic estimation process of oscillator velocity and external force parameters for the mathematical pendulum model are presented.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"53 31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503526","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}

{"title":"Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector","authors":"Mykola Krasnoshchok","doi":"10.37069/1683-4720-2023-37-3","DOIUrl":"https://doi.org/10.37069/1683-4720-2023-37-3","url":null,"abstract":"We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_mu(s_theta$ under some conditions on weight $mu$, higher order of derivatives $k$ and the opening of the angle $theta$.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503531","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}

{"title":"On the possibility of joining two pairs of points in convex domains using paths","authors":"Oleksandr Dovhopiaty","doi":"10.37069/10.37069/1683-4720-2023-37-1","DOIUrl":"https://doi.org/10.37069/10.37069/1683-4720-2023-37-1","url":null,"abstract":"This article is devoted to the possibility of joining of two pairs of points of a convex domains by curves. We are interested in the case when these curves are not farther from each other than the distance between their end points, possibly, up to some absolute multiplicative constant. We have obtained some upper and lower bounds for the modulus of families of paths joining curves mentioned above.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503534","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}

{"title":"Pointwise estimates of solutions to weighted parabolic p-Laplacian equation via Wolff potential","authors":"Yevhen Zozulia","doi":"10.37069/1683-4720-2022-36-07","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-07","url":null,"abstract":"For the weighted parabolic equation vleft(x right)u_{t} -{hbox{div}({w(x)| nabla u |^{p-2}}} nabla u) = f , p >{2} we prove the local boundedness for weak solutions in terms of the weighted Wolff potential of the right-hand side of equation.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996254","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}

{"title":"Identification of parameters of non-linear oscillators","authors":"Nadiya Zhogoleva, Volodymyr Shcherbak","doi":"10.37069/1683-4720-2022-36-06","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-06","url":null,"abstract":"Many applied control problems are characterized by a situation where some or all parameters of the initial dynamic system are unknown. In such cases, the problem of identification arises, which consists in determining the unknown parameters of the system based on information about its output - known information about movement. The ability to solve the problem of identification is an essential property of identifiability depends on the analytical structure of the right-hand sides of the dynamics equations and available information [1]. To solve the identification problem itself, this work uses the method of invariant relations [2], which was developed in analytical mechanics and is intended, in particular, for finding partial solutions (dependencies between variables) in problems of the dynamics of a rigid body with a fixed point. The modification of this method to the problems of the theory of control, observation made it possible to synthesize additional connections between the known and unknown quantities of the original system that arise during the movement of its extended model [3 - 5]. It is worth noting that a some more general approach, which forms a suitable method for solving observation problems for nonlinear dynamic systems due to the synthesis of an invariant manifold in the space of an extended system, was proposed in the works [6], [7] as a certain modification of the method stabilization of nonlinear systems I&I (Input and Invariance). The purpose of this work is to spread the method of synthesis of invariant relations in control problems to the problem of identifying parameters of pendulum systems. A general scheme for constructing asymptotically accurate estimates of the parmeters of a two-dimensional dynamical system is proposed. A relatively simple case of the identification problem will be considered, namely: 1) the output of the original system is the complete phase vector and 2) the system depends linearly on the unknown parameters. Generalizations to more general designs of input-output systems, including with the involvement of information about the output obtained on several trajectories, can be carried out using the approach described below and is the subject of a separate study. The computational experiment on the estimation of the parameters of the mathematical pendulum confirms the efficiency of the proposed identification scheme.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996411","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}

{"title":"Influence of dissipative asymmetry on the of rotation stability in a resisting medium of a asymmetric rigid body under the action of a constant moment in inertial reference frame","authors":"Yuriy Kononov, Akram Cheib","doi":"10.37069/1683-4720-2022-36-08","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-08","url":null,"abstract":"Assuming that the center of mass of an asymmetric rigid body is located on the third main axis of inertia of a rigid body, the influence of dissipative asymmetry on the stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body is estimated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are represented by a system of three inequalities. The first and second inequalities have the first degree with respect to the dissipative asymmetry, and the third inequality has the third degree. The third inequality is the most difficult to study. Analytical studies of the influence of small and large dissipative asymmetries, restoring, overturning and constant moments on the stability of rotation of a rigid body are carried out. Conditions for asymptotic stability are obtained for sufficiently small values of the dissipative asymmetry and conditions for instability for sufficiently large values of the asymmetry. The stability conditions are written down to the second order of smallness with respect to the constant moment and the first - with respect to the restoring or overturning moments. Stability conditions for the rotation of a rigid body around the center of mass are studied.","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996255","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}

{"title":"Observer of harmonic oscillator parameters","authors":"Volodymyr Shcherbak","doi":"10.37069/1683-4720-2022-36-11","DOIUrl":"https://doi.org/10.37069/1683-4720-2022-36-11","url":null,"abstract":"This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9], [10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].","PeriodicalId":484640,"journal":{"name":"Trudy Instituta prikladnoj matematiki i mehaniki","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996257","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}