We prove the lower bounds for the functions introduced as the maximal constants in the Hardy and Rellich type inequalities for polyharmonic operator of order m in domains in a Euclidean space. In the proofs we employ essentially the known integral inequality by O.A. Ladyzhenskaya and its generalizations. For the convex domains we establish two generalizations of the known results obtained in the paper M.P. Owen, Proc. Royal Soc. Edinburgh, 1999 and in the book A.A. Balinsky, W.D. Evans, R.T. Lewis, The analysis and geometry of hardy’s inequality, Springer, 2015. In particular, we obtain a new proof of the theorem by M.P. Owen for polyharmonic operators in convex domains. For the case of arbitrary domains we prove universal lower bound for the constants in the inequalities for mth order polyharmonic operators by using the products of m different constants in Hardy type inequalities. This allows us to obtain explicit lower bounds for the constants in Rellich type inequalities for the dimension two and three. In the last section of the paper we discuss two open problems. One of them is similar to the problem by E.B. Davies on the upper bounds for the Hardy constants. The other problem concerns the comparison of the constants in Hardy and Rellich type inequalities for the operators defined in three-dimensional domains.
我们证明了欧几里德空间中m阶多谐算子的Hardy和Rellich型不等式中以极大常数形式引入的函数的下界。在证明中，我们基本上采用了Ladyzhenskaya提出的已知的积分不等式及其推广。对于凸域，我们建立了M.P. Owen, Proc. Royal Soc论文中已知结果的两个推广。A.A. Balinsky, W.D. Evans, R.T. Lewis，《哈代不等式的分析与几何》，Springer, 2015。特别地，我们得到了M.P. Owen关于凸域上多谐算子定理的一个新的证明。在任意定域下，利用Hardy型不等式中m个不同常数的积证明了m阶多调和算子不等式中常数的普遍下界。这使我们能够获得第2维和第3维Rellich型不等式中常数的显式下界。在论文的最后一部分，我们讨论了两个开放的问题。其中一个类似于E.B. Davies关于Hardy常数上界的问题。另一个问题涉及在三维域上定义的算子的Hardy型不等式和Rellich型不等式常数的比较。
{"title":"Estimates of Hardy - Rellich constants for polyharmonic operators and their generalizations","authors":"F. Avkhadiev","doi":"10.13108/2017-9-3-8","DOIUrl":"https://doi.org/10.13108/2017-9-3-8","url":null,"abstract":"We prove the lower bounds for the functions introduced as the maximal constants in the Hardy and Rellich type inequalities for polyharmonic operator of order m in domains in a Euclidean space. In the proofs we employ essentially the known integral inequality by O.A. Ladyzhenskaya and its generalizations. For the convex domains we establish two generalizations of the known results obtained in the paper M.P. Owen, Proc. Royal Soc. Edinburgh, 1999 and in the book A.A. Balinsky, W.D. Evans, R.T. Lewis, The analysis and geometry of hardy’s inequality, Springer, 2015. In particular, we obtain a new proof of the theorem by M.P. Owen for polyharmonic operators in convex domains. For the case of arbitrary domains we prove universal lower bound for the constants in the inequalities for mth order polyharmonic operators by using the products of m different constants in Hardy type inequalities. This allows us to obtain explicit lower bounds for the constants in Rellich type inequalities for the dimension two and three. In the last section of the paper we discuss two open problems. One of them is similar to the problem by E.B. Davies on the upper bounds for the Hardy constants. The other problem concerns the comparison of the constants in Hardy and Rellich type inequalities for the operators defined in three-dimensional domains.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72533479","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}
. In the work we study an analogue of Tricomi equation for a third order parabolic-hyperbolic equation with smaller derivatives having multiple characteristics. Under certain conditions for the given functions and parameters involved in the considered equation, we prove unique solvability theorem for the studied problem. The uniqueness of the solution is proved by means of the generalized Tricomi method, while the existence is proved via the method of integral equations.
{"title":"Dirichlet boundary value problem for a third order parabolic-hyperbolic equation with degenerating type and order in the hyperbolicity domain","authors":"Zh.A. Balkizov","doi":"10.13108/2017-9-2-25","DOIUrl":"https://doi.org/10.13108/2017-9-2-25","url":null,"abstract":". In the work we study an analogue of Tricomi equation for a third order parabolic-hyperbolic equation with smaller derivatives having multiple characteristics. Under certain conditions for the given functions and parameters involved in the considered equation, we prove unique solvability theorem for the studied problem. The uniqueness of the solution is proved by means of the generalized Tricomi method, while the existence is proved via the method of integral equations.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83778728","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}
. We consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the higher derivative and a large gradient of the initial function. By means of the numerical and analytic methods we show that the formal asymptotics obtained by renormalization is an asymptotic solution to the KdV equation. We obtain the graphs of the asymptotic solutions including the case of non-monotone initial data.
{"title":"Modelling compression waves with a large initial gradient in the Korteweg - de Vries hydrodynamics","authors":"S. Zakharov, A. E. El’bert","doi":"10.13108/2017-9-1-41","DOIUrl":"https://doi.org/10.13108/2017-9-1-41","url":null,"abstract":". We consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the higher derivative and a large gradient of the initial function. By means of the numerical and analytic methods we show that the formal asymptotics obtained by renormalization is an asymptotic solution to the KdV equation. We obtain the graphs of the asymptotic solutions including the case of non-monotone initial data.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87496518","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}
For the most known differential substitutions relating scalar evolution equations, the sets of the equations admitting them consist of not finitely many equations but they form families parametrized by an arbitrary function. Some differential substitutions for evolution systems also have a similar property. In the present paper we obtain necessary and sufficient conditions for a differential substitution to be admitted by a family of evolution systems depending on an arbitrary function. We also give explicit formulae for finding the corresponding family of evolution systems in the case when these conditions are satisfied. As an example, the family of systems admitting a multi-component Cole-Hopf substitution is constructed. We demonstrate that this family contains all linear systems, whose right hand sides contain no terms independent of the derivatives. As a result, we obtain a set of C-integrable systems of arbitrary high order. Another example considered in the paper is a multi-component analogue of the substitution v = ux + exp(u). We show that this multi-component substitution is also admitted by a family of evolution systems depending on an arbitrary function.
{"title":"On differential substitutions for evolution systems","authors":"S. Startsev","doi":"10.13108/2017-9-4-108","DOIUrl":"https://doi.org/10.13108/2017-9-4-108","url":null,"abstract":"For the most known differential substitutions relating scalar evolution equations, the sets of the equations admitting them consist of not finitely many equations but they form families parametrized by an arbitrary function. Some differential substitutions for evolution systems also have a similar property. In the present paper we obtain necessary and sufficient conditions for a differential substitution to be admitted by a family of evolution systems depending on an arbitrary function. We also give explicit formulae for finding the corresponding family of evolution systems in the case when these conditions are satisfied. As an example, the family of systems admitting a multi-component Cole-Hopf substitution is constructed. We demonstrate that this family contains all linear systems, whose right hand sides contain no terms independent of the derivatives. As a result, we obtain a set of C-integrable systems of arbitrary high order. Another example considered in the paper is a multi-component analogue of the substitution v = ux + exp(u). We show that this multi-component substitution is also admitted by a family of evolution systems depending on an arbitrary function.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88910853","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}
A generic scheme based on the matrix Riemann-Hilbert problem theory is proposed for constructing classical special functions satisfying difference equations. These functions comprise gammaand zeta functions, as well as orthogonal polynomials with corresponding recurrence relations. We show that all difference equations are the compatibility conditions of certain Lax pair coming from the Riemann-Hilbert problem. At that, the integral representations for solutions to the classical Riemann-Hilbert problem on duality of analytic functions on a contour in the complex plane are generalized for the case of discrete measures, that is, for infinite sequences of points in the complex plane. We establish that such generalization allows one to treat a series of nonlinear difference equations integrable in the sense of solitons theory. The solutions to the mentioned Riemann-Hilbert problems allows us to reproduce analytic properties of classical special functions described in handbooks and to describe a series of new functions pretending to be special. For instance, this is true for difference Painlevé equations. We provide the example of applying a difference second type Painlevé equation to the representation problem for a symmetric group. Mathematics Subject Classification: 33C05, 33C12, 34M55, 34M40, 34E20, 34M60 In work [18], there was considered a scheme for describing classical special functions based on the matrix Riemann-Hilbert problem. It was shown that such functions satisfying ordinary differential equations can be represented in terms of a solution to some Riemann-Hilbert problem, that is, in terms of the problem on recovering an analytic function by its boundary values. In this way, for the corresponding differential equations, there was checked the integrability property treated in the sense of the solutions theory [1], [26]. Such treating of the integrability property as calculating of the values of a function by its global behavior means the presence of an integrable representation for this function. In fact, the method of the Riemann-Hilbert problem demonstrates the equivalency of these two definitions of the integrability [6], [15]. The functions covered by such treating of the integrability are, for instance, hypergeometric and elliptic functions. However, in the handbooks, see, for instance, [7], [14], [27], there are other special functions satisfying no differential equations. Among such functions are Gamma and zeta functions and their generalizations arising in the number theory, combinatorics and the groups representation theory. How one can extend the method of the Riemann-Hilbert problem to these special functions? In the present paper we attempt to answer this question. The key point is that there exists a discrete equation satisfied by special functions. It turns out that these equations can be treated within the scheme of the solitons theory. Namely, for each discrete equation we provide the Lax pair of two linear equations and
{"title":"Discrete integrable equations and special functions","authors":"Victor Yur'evich Novokshenov","doi":"10.13108/2017-9-3-118","DOIUrl":"https://doi.org/10.13108/2017-9-3-118","url":null,"abstract":"A generic scheme based on the matrix Riemann-Hilbert problem theory is proposed for constructing classical special functions satisfying difference equations. These functions comprise gammaand zeta functions, as well as orthogonal polynomials with corresponding recurrence relations. We show that all difference equations are the compatibility conditions of certain Lax pair coming from the Riemann-Hilbert problem. At that, the integral representations for solutions to the classical Riemann-Hilbert problem on duality of analytic functions on a contour in the complex plane are generalized for the case of discrete measures, that is, for infinite sequences of points in the complex plane. We establish that such generalization allows one to treat a series of nonlinear difference equations integrable in the sense of solitons theory. The solutions to the mentioned Riemann-Hilbert problems allows us to reproduce analytic properties of classical special functions described in handbooks and to describe a series of new functions pretending to be special. For instance, this is true for difference Painlevé equations. We provide the example of applying a difference second type Painlevé equation to the representation problem for a symmetric group. Mathematics Subject Classification: 33C05, 33C12, 34M55, 34M40, 34E20, 34M60 In work [18], there was considered a scheme for describing classical special functions based on the matrix Riemann-Hilbert problem. It was shown that such functions satisfying ordinary differential equations can be represented in terms of a solution to some Riemann-Hilbert problem, that is, in terms of the problem on recovering an analytic function by its boundary values. In this way, for the corresponding differential equations, there was checked the integrability property treated in the sense of the solutions theory [1], [26]. Such treating of the integrability property as calculating of the values of a function by its global behavior means the presence of an integrable representation for this function. In fact, the method of the Riemann-Hilbert problem demonstrates the equivalency of these two definitions of the integrability [6], [15]. The functions covered by such treating of the integrability are, for instance, hypergeometric and elliptic functions. However, in the handbooks, see, for instance, [7], [14], [27], there are other special functions satisfying no differential equations. Among such functions are Gamma and zeta functions and their generalizations arising in the number theory, combinatorics and the groups representation theory. How one can extend the method of the Riemann-Hilbert problem to these special functions? In the present paper we attempt to answer this question. The key point is that there exists a discrete equation satisfied by special functions. It turns out that these equations can be treated within the scheme of the solitons theory. Namely, for each discrete equation we provide the Lax pair of two linear equations and ","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86082925","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}
We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basis for a space by the bases in subspaces. We also consider the cases when the bases for subspaces are isomorphic and the corresponding isomorphisms are not needed. The completeness, minimality and uniform minimality of the corresponding systems are studied. This approach has extensive applications in the spectral theory of discontinuous differential operators.
{"title":"On basicity of eigenfunctions of second order discontinuous differential operator","authors":"B. Bilalov, T. Gasymov","doi":"10.13108/2017-9-1-109","DOIUrl":"https://doi.org/10.13108/2017-9-1-109","url":null,"abstract":"We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basis for a space by the bases in subspaces. We also consider the cases when the bases for subspaces are isomorphic and the corresponding isomorphisms are not needed. The completeness, minimality and uniform minimality of the corresponding systems are studied. This approach has extensive applications in the spectral theory of discontinuous differential operators.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75688097","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}
We consider a canonical factorization and integral representation for the derivatives of the conformal mappings of circular domains on finitely-connected non-Smirnov type domains. By means of the functions in the Zygmund class, we obtain the conditions for the global univalence. Earlier similar results were obtained by a series of authors just for simply-connected domains.
{"title":"Conformal mappings of circular domains on finitely-connected non-Smirnov type domains","authors":"F. Avkhadiev, P. Shabalin","doi":"10.13108/2017-9-1-3","DOIUrl":"https://doi.org/10.13108/2017-9-1-3","url":null,"abstract":"We consider a canonical factorization and integral representation for the derivatives of the conformal mappings of circular domains on finitely-connected non-Smirnov type domains. By means of the functions in the Zygmund class, we obtain the conditions for the global univalence. Earlier similar results were obtained by a series of authors just for simply-connected domains.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72495791","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}
We propose an algorithm for integrating n-th order ordinary differential equations (ODE) admitting n-dimensional Lie algebras of operators. The algorithm is based on invariant representation of the equations by the invariants of the admitted Lie algebra and application of an operator of invariant differentiation of special type. We show that in the case of scalar equations this method is equivalent to the known order reduction methods. We study an applicability of the suggested algorithm to the systems of m kth order ODEs admitting km-dimensional Lie algebras of operators. For the admitted Lie algebra we obtain a condition ensuring the possibility to construct the operator of invariant differentiation of a special type and to reduce the order of the considered system of ODEs. This condition is the implication of the existence of nontrivial solutions to the systems of linear algebraic equations, where the coefficients are the structural constants of the Lie algebra. We present an algorithm for constructing the (km − 1)-dimensional Lie algebra for the reduced system. The suggested approach is applied for integrating the systems of two second order equations.
{"title":"Operator of invariant differentiation and its application for integrating systems of ordinary differential equations","authors":"R. Gazizov, A. Gainetdinova","doi":"10.13108/2017-9-4-12","DOIUrl":"https://doi.org/10.13108/2017-9-4-12","url":null,"abstract":"We propose an algorithm for integrating n-th order ordinary differential equations (ODE) admitting n-dimensional Lie algebras of operators. The algorithm is based on invariant representation of the equations by the invariants of the admitted Lie algebra and application of an operator of invariant differentiation of special type. We show that in the case of scalar equations this method is equivalent to the known order reduction methods. We study an applicability of the suggested algorithm to the systems of m kth order ODEs admitting km-dimensional Lie algebras of operators. For the admitted Lie algebra we obtain a condition ensuring the possibility to construct the operator of invariant differentiation of a special type and to reduce the order of the considered system of ODEs. This condition is the implication of the existence of nontrivial solutions to the systems of linear algebraic equations, where the coefficients are the structural constants of the Lie algebra. We present an algorithm for constructing the (km − 1)-dimensional Lie algebra for the reduced system. The suggested approach is applied for integrating the systems of two second order equations.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84721430","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}
The work is devoted to the substantial extension of the space of the potentials in the inverse scattering problem for the linear Schrödinger equation on the real axis. We consider the Schrödinger operator with a potential in the space of generalized functions. This extension includes not only the potential like delta function, but also exotic cases like Cantor functions. In this way we establish the conditions on existence and uniqueness of Jost solutions. We study their analytic properties. We provide some estimates for the Jost solutions and their derivatives. We show that the Schrödinger equation with the distribution potential can be uniformly approximated by the equations with smooth potentials.
{"title":"Some properties of Jost functions for Schrödinger equation with distribution potential","authors":"R. Kulaev, A. Shabat","doi":"10.13108/2017-9-4-59","DOIUrl":"https://doi.org/10.13108/2017-9-4-59","url":null,"abstract":"The work is devoted to the substantial extension of the space of the potentials in the inverse scattering problem for the linear Schrödinger equation on the real axis. We consider the Schrödinger operator with a potential in the space of generalized functions. This extension includes not only the potential like delta function, but also exotic cases like Cantor functions. In this way we establish the conditions on existence and uniqueness of Jost solutions. We study their analytic properties. We provide some estimates for the Jost solutions and their derivatives. We show that the Schrödinger equation with the distribution potential can be uniformly approximated by the equations with smooth potentials.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77910318","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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