Order Differential Operators Research Articles

A Sequence of Weighted Birman–Hardy–Rellich Inequalities with Logarithmic Refinements

The principal aim of this paper is to extend Birman’s sequence of integral inequalities originally obtained in Mat. Sb. (N.S.) 55(97), 125–174, (1961), and containing Hardy’s and Rellich’s inequality as special cases, to a sequence of inequalities that incorporates power weights \(x^{\alpha }\) for x varying in intervals \((0,\rho ), \rho \in (0,\infty ) \cup \{\infty \}\), on either side and logarithmic refinements on the right-hand side of the inequality as well. Employing a new technique of proof relying on a combination of transforms originally due to Hartman and Müller-Pfeiffer, the parameter \(\alpha \in {{\mathbb {R}}}\) in the power weights is now unrestricted, considerably improving on prior results in the literature. We also discuss optimality of the constants in these inequalities. This continues a tradition of logarithmic refinements in connection with Hardy’s inequality, going back to work in oscillation theory by Kneser (Math. Ann. 42, 409–435, (1893)), Hartman (Am. J. Math. 70, 764-779 (1948)), Hille (Trans. Am. Math. Soc. 64, 234-252 (1948)), and Rellich (Math. Ann. 122, 343–368 (1951)), resulting in a sequence of sharp statements of boundedness from below by zero of a class of homogeneous 2mth order differential operators on \(C_0^{\infty }((0,\rho ))\). We also prove the analogous inequalities on exterior intervals, that is, for \(f \in C_0^{\infty }((\rho ,\infty ))\). Finally, we also indicate a vector-valued version of these inequalities, replacing complex-valued \(f(\,\cdot \,)\) by \(f(\,\cdot \,) \in {{\mathcal {H}}}\), with \({{\mathcal {H}}}\) a complex, separable Hilbert space.

Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\\colon L^2(\\mathbb{R})\\to L^2(\\mathbb{R})$ have the form $$ P:=-\\frac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\\mathbf{1}{(-\\infty,\\lambda^2]}(P)$ has a full asymptotic expansion in powers of $\\lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with \_dense pure point spectrum. The proof combines the gauge transform methods of Parnovski–Shterenberg and Sobolev with Melrose's scattering calculus.

Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators

This work is a galoisian study of the spectral problem $L\Psi=\lambda\Psi$, for algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $\lambda$ an algebraic variable over $C$, forcing the consideration of a new of coefficients for $L-\lambda$, whose of constants is the $C(\Gamma)$ of the spectral curve $\Gamma$. Since $C(\Gamma)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over $\Gamma$, called Spectral Picard-Vessiot field of $L-\lambda$. An existence theorem is proved using differential algebra, allowing to recover classical Picard-Vessiot theory for each $ \lambda = \lambda_0 $. For rational spectral curves, the appropriate algebraic setting is established to solve $L\Psi=\lambda\Psi$ analitically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

Orthogonal polynomials and diffusion operators

We want to describe the triplets (\Omega, (g), \mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain \Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(\mu), and \mu is some admissible measure. Up to affine transformation, we find 11 compact domains in dimension 2, and also give some non--compact cases in this dimension.

On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m

Abstract The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m {\Omega\subset\mathbb{R}^{m}} with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ ( 1 + α , Γ ) {\operatorname{Lip}(1+\alpha,\Gamma)} .

Direct exchange calculation for unstructured micromagnetic meshes

In micromagnetic simulations it is necessary to compute the effective field associated to the exchange interaction, which corresponds to applying the Laplace operator to the magnetization field. The most widely used approach for computing the exchange field is to apply finite-difference schemes to homogeneous Cartesian meshes. In many situations it would be computationally advantageous to perform micromagnetic simulation on inhomogeneous meshes, but doing so presents a challenge with regard to the calculation of the exchange field. Here we present the construction of an exchange field calculation method for inhomogeneous meshes. A novel second order differential operator method is introduced and applied to a series of analytical functions on irregular meshes and compared to existing methods, finding an accuracy improvement of up to 81% relative to the second order finite difference method. The method is then used in combination with the MagTense framework and tested on the μmag standard micromagnetics problems 3 and 4. Here we find that especially for tetrahedral meshes, the proposed method results in much faster convergence as function of mesh size. We demonstrate that the method converges stably even for meshes where some mesh elements have a volume as small as 1/8 compared to the base resolution. Finally, we show that the method works equally well for meshes with spatially varying material parameters.

Convolution-Based Model-Solving Method for Three-Dimensional, Unsteady, Partial Differential Equations

Neural networks are increasingly used widely in the solution of partial differential equations (PDEs). This letter proposes 3D-PDE-Net to solve the three-dimensional PDE. We give a mathematical derivation of a three-dimensional convolution kernel that can approximate any order differential operator within the range of expressing ability and then conduct 3D-PDE-Net based on this theory. An optimum network is obtained by minimizing the normalized mean square error (NMSE) of training data, and L-BFGS is the optimized algorithm of second-order precision. Numerical experimental results show that 3D-PDE-Net can achieve the solution with good accuracy using few training samples, and it is of highly significant in solving linear and nonlinear unsteady PDEs.

Complex symmetric weighted composition-differentiation operators of order $n$ on the weighted Bergman spaces

We study the complex symmetric structure of weighted composition--differentiation operators of order $n $ on the weighted Bergman spaces $A_{\alpha}^2$ with respect to some conjugations. Then we provide some examples of these operators.

Local density peaks clustering with small size distance matrix

Clustering by using density peaks (DPC) is a powerful clustering method. The cluster centers in DPC are characterized by a higher density than their neighbors and by a relatively large distance from points with higher densities, which leads to the storage of a distance matrix with the same size of sample size. In this paper, to deduce the storage of the distance matrix, we propose a local density peaks clustering (LDPC) by introducing the local region by reverse nearest neighbors. Compared with DPC, the potential cluster centers and the potential haloes are defined, and the potential cluster centers distance matrix is introduced to replace the storage of the sample distance matrix, which makes it more practical for large scale data. At the same time, the local information is used to construct the high densities, leads to more robust to the data structure. Further, the second order differential operator is used to give the number of clustering automatically. Preliminary computational results demonstrate the effectiveness of proposed LDPC over DPC on artificial datasets and face image considered.

The second regularized trace of even order differential operators with operator coefficient

In this paper, we investigate the spectrum of the self adjoint operator L defined by L := (-1)r d2r/dx2r + A + Q(x), where A is a self adjoint operator, Q(x) is a nuclear operator in a separable Hilbert space, and we derive asymptotic formulas for the sum of eigenvalues of the operator L.

A Generalized Method of Undetermined Coefficients

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples.

Distribution dependent SDEs for Navier-Stokes type equations

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

<abstract><p>In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.</p></abstract>

Modeling the Virus Infection at the Population Level.

As pointed out by many researchers in the last few decades, differential equations with fractional (non-integer) order differential operators, in comparison with classical integer order ones, have apparent advantages in modeling. A Caputo fractional order system of ordinary differential equations is introduced to model the virus infection at the population level in this chapter. As well known, there are two main methods to study the dynamics of a model: qualitative analysis and numerical modeling. Here the qualitative analysis, including uniqueness, invariant set, and stability, is first presented with intuitive derivation. Then the famous genetic algorithm is introduced to numerically model the dynamics of virus infection, i.e. to adjust the parameters of the Caputo fractional model such that its solution can properly fit real data and predict future.

Advanced fibonacci sequence and its sum by second order variable co-efficient difference operator

We introduce a second order difference operator with specific powers of variable co-efficient and its inverse in this study, which allows us to derive the (α1tr1, α2tr2 )-Fibonacci sequence and its summation. This series is known as the Fibonacci sequence with variable co-efficients (VCFS). On the sum of the terms of the variable co-efficient Fibonacci sequence, some theorems and intriguing findings are generated. To demonstrate our findings, appropriate instances arepresented.

New examples of Krall–Meixner and Krall–Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials

We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure ρa,c and Hahn measure ρa,b,N by removing a finite number of their mass points when the parameter c of the Meixner measure and a or b of the Hahn measure are positive integers. From the new Krall–Meixner families we construct new families of exceptional Meixner and Laguerre polynomials.

Unique continuation property of solutions to general second order elliptic systems

Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

Carleman estimates and boundedness of associated multiplier operators

Let P(D) be the Laplacian or the wave operator □. The following type of Carleman estimate is known to be true on a certain range of p, q: with C independent of The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge [1] and Jeong-Kwon-Lee [2]. The range of p, q for which the uniform Sobolev type estimates hold was completely characterized for the second order differential operators with nondegenerate principal part. But the optimal range of p, q for which the Carleman estimate holds has not been clarified before. When □, or the heat operator, we obtain a complete characterization of the admissible p, q for the aforementioned type of Carleman estimate. For this purpose we investigate Lp –Lq boundedness of related multiplier operators. As applications, we also obtain some unique continuation results.

Analyticity of resolvents of elliptic operators on quantum graphs with small edges

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to ε, where ε is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator Hε with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in ε. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of Hε are analytic in ε. This allows us to represent the resolvent of Hε by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of ε. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in ε-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic ε-dependence of the coefficients in the differential expression and vertex conditions.

Curl-free positive definite form of time-harmonic Maxwell’s equations well-suitable for iterative numerical solving

A new form of time-harmonic Maxwell’s equations is developed on the base of the standard ones and proposed for numerical modeling. It is written for the magnetic field strength H, electric displacement D, vector potential A and the scalar potential Φ. There are several attractive features of this form. The 1st one is that the differential operator acting on these quantities is positive. The 2nd is absence of curl operators among the leading order differential operators. The Laplacian stands for leading order operator in the equations for H, A and Φ, while the gradient of divergence stands for D. The 3rd feature is absence of space varied coefficients in the leading order differential operators that provides diagonal domination of the resulting matrix of the discretized equations. A simple example is given to demonstrate the applicability of this new form of time-harmonic Maxwell’s equations.