Order Linear Differential Operator Research Articles

Graded jet geometry

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of Z-graded manifolds and vector bundles.Our aim is to construct the k-th order jet bundle JEk of an arbitrary Z-graded vector bundle E over an arbitrary Z-graded manifold M. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of k-th order (linear) differential operators DEk on E. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric k-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of Z-graded vector bundles over Z-graded manifolds are recalled.

Method of fundamental solutions: New approximation results and applications

The approximation properties of shifted fundamental solutions are stated for a general class of second order linear elliptic differential operators. Motivated by the method of the fundamental solutions, the singularities of the fundamental solutions are located in the vicinity of a fixed surface, which is the boundary of a 3-dimensional Lipschitz domain. The approximation is estimated first in Ck-norm. The corresponding convergence rate is determined depending on the density of the singularities. This delivers a solid theoretical basis for the method of fundamental solutions. The results are extended to exterior domains and as a new application, the numerical solution of Dirichlet-to-Neumann problems are constructed and analyzed on unbounded domains. The efficiency of this new method is demonstrated in series of numerical experiments.

SECOND ORDER DIFFERENTIAL OPERATORS WITH ALGEBRAIC SOLUTIONS

We are surveying recent results that describe second order differential operators having only algebraic solutions in the sense of Galois theory. We call such operators algebraic. For hypergeometric operators, this problem was studied by Schwarz and Klein who also gave results that describe all second order linear differential operators with a full set of algebraic solutions. Starting from their work, we see algebraic operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We are surveying some of the main results describing second order operators with a full set of algebraic solutions, especially those obtained by using the properties of the pull-back functions. Using the Grothendieck correspondence, these properties transfer to properties for their corresponding dessins d’enfants.

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

We study integrally bounded solutions of the differential equation A(x)=z, where A is a linear differential operator of order l defined on functions x:R→H (R=(−∞,∞), H () and H is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ0. Conditions are given on the operator A and the function ψ:R→H that guarantee an inverse inequality of the following form for the solutions x under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant is independent of the choice of a real number t and function x.

Best Constant in Ulam Stability for the Third Order Linear Differential Operator with Constant Coefficients

The authors of the present paper previously proved the Ulam stability for the n-th-order linear differential operator with constant coefficients. They obtained its best Ulam constant for the case of distinct roots of the characteristic equation. However, a complete answer to the problem of the best Ulam constant was later obtained only for the second-order linear differential operator. This paper deals with the Ulam stability of the third-order linear differential operator with constant coefficients acting in a Banach space. The paper’s main purpose is to obtain the best Ulam constant of this operator, thus completing the previous research in the field.

An asymptotic expansion of eigenpolynomials for a class of linear differential operators

AbstractConsider an M‐th order linear differential operator, ,where is a monic complex polynomial such that and are complex polynomials such that . It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure μ. We obtain an asymptotic expansion of the eigenpolynomials of in compact subsets out of the support of μ. In particular, we solve a conjecture posed in Masson and Shapiro [On polynomial eigenfunctions of a hypergeometric type operator. Exper Math. 2001;10:609‐618].

Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) → F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 → F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 ο D = 0 implies ad (D) ο ad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Fast algorithms for interpolation with L-splines for differential operators [formula omitted] of order 4 with constant coefficients

In the classical theory of cubic interpolation splines there exists an algorithm which works with only On arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with On arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of L-splines where L is a linear differential operator of order 4 with constant coefficients. Criteria are given such that the associated matrix R is strictly diagonally dominant which implies the existence of a fast algorithm for interpolation. We provide an example of two L-splines interpolating S&P500 data.

Sobolev regularity for linear growth functionals acting on ℂ-elliptic operators

Abstract In this paper, we prove the higher Sobolev regularity of minimizers for convex integral functionals evaluated on linear differential operators of order one. This work intends to generalize the already existing theory for the cases of full and symmetric gradients to the entire class of ${\mathbb C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.

L-Spline Interpolation for Differential Operators of Order 4 with Constant Coefficients


 
 
 In this paper it is shown that many features from polynomial spline methods used in nonparametric regression and smoothing procedures carry over to the class of L-splines where L is a linear differential operator of order 4 with constant coefficients. Special attention is given to the question whether an analogue of the Reinsch algorithm is valid and criteria are given such that the associated matrix R is strictly diagonal dominant
 
 

Reduced limit approach to semilinear partial differential equations (PDEs) involving the fractional Laplacian with measure data

We study the following partial differential equation (PDE) \begin{align} \begin{split} (-\Delta)^s u + g(x,u) & = \mu\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\,\mbox{in}\,\,\mathbb{R}^N\setminus\Omega,\label{eqn_abs} \end{split} \end{align} where $(-\Delta)^s$ is the fractional Laplacian operator, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $\partial\Omega$ being the boundary of $\Omega$, $g(.,.)$ is a nonlinear function defined over $\Omega\times\mathbb{R}$. Let $(\mu_n)_n$ be a sequence of measure in $\Omega$. Assume that there exists a solution $u_n$ with data $\mu_n$, i.e. $u_n$ satisfies the equation (0.1) with $\mu=\mu_n$. We further assume that the sequence of measures weakly converges to $\mu$, while $(u_n)_n$ converges to $u$ in $L^1(\Omega)$. In general, $u$ is not a solution to the partial differential equation in (0.1) with datum $(\mu,0)$. However, there exists a measure $\mu^{\#}$ such that $u$ is a solution of the partial differential equation with this data. $\mu^{\#}$ is called the reduced limit of the sequence $(\mu_n)_n$. We investigate the relation between weak limit $\mu$ and the reduced limit $\mu^{\#}$ and the dependence of $\mu^{\#}$ to the sequence $(\mu_n)_n$. A closely related problem was studied by Bhakta and Marcus [3] and then by Giri and Choudhuri [15] but for the case of a Laplacian and a general second order linear elliptic differential operator, respectively instead of a fractional Laplacian.

Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is studied and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.

On local continuous solvability of equations associated to elliptic and canceling linear differential operators

Consider A(x,D):C∞(Ω,E)→C∞(Ω,F) an elliptic and canceling linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a finite dimension complex vector space E to a finite dimension complex vector space F and A⁎(x,D) its adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation A⁎(x,D)v=f (in the distribution sense) for a given distribution f; more precisely we show that any x0∈Ω is contained in a neighborhood U⊂Ω in which its continuous solvability is characterized by the following condition on f: for every ε>0 and any compact set K⊂⊂U, there exists θ=θ(K,ε)>0 such that the following holds for all smooth function φ supported in K:|f(φ)|⩽θ‖φ‖Wν−1,1+ε‖A(x,D)φ‖L1, where Wν−1,1 stands for the homogenous Sobolev space of all L1 functions whose derivatives of order ν−1 belongs to L1(U).This characterization implies and extends results obtained before for operators associated to elliptic complexes of vector fields (see [1]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator by Bourgain and Brezis in [2] and by De Pauw and Pfeffer in [3].

Invariants of Forth Order Linear Differential Operators

In this paper, we study scalar the forth order linear differential operators over an oriented 2-dimensional manifold. We investigate differential invariants of these operators and show their application to the equivalence problem.

Second Order Linear Differential Operators over High Rank Tensor Fields

Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered in detail. Square matrices are introduced, consisting of differential operators $${\text{Di}}{{{\text{v}}}_{{(l)}}}{\text{Gra}}{{{\text{d}}}_{{(k)}}}$$ , $${\text{Gra}}{{{\text{d}}}_{{(k)}}}{\text{Di}}{{{\text{v}}}_{{(l)}}}$$ , and their relationship is established. An explicit expression is written for the repeated operator rotor. All introduced generalized operators in particular cases agree in their properties with the corresponding classical operators in vector and tensor analysis.

On essential self-adjointness for first order differential operators on domains in $\mathbb R^d$

We consider general symmetric systems of first order linear partial differential operators on domains \Omega \subset \mathbb R^d , and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to C^1 (\Omega) and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for certain wave propagation problems of classical physics.

Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painlevé equations: A Riemann–Hilbert problem perspective

In this paper we use the Riemann–Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials on the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first degree matrix polynomials, is given. All of these are applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because of the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of a hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete Painlevé I equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painlevé I equation is found.

Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators

Let A(x,D) be an elliptic linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a complex vector space E to a complex vector space F. In this paper we show that if ℓ∈R satisfies 0<ℓ<N and ℓ≤ν, then the estimate(∫RN|Pν−ℓ(x,D)u(x)|q|x|−N+(N−ℓ)qdx)1/q≤C‖A(x,D)u‖L1 holds locally for every u∈Cc∞(U;E) and 1≤q<N/(N−ℓ) assuming A(x,D) is canceling, i.e. ⋂ξ∈RN∖{0}A(x0,ξ)[E]={0} for each x0∈Ω. Here Pν−ℓ(x,D) is a properly supported pseudo-differential operator in Hörmander's class OpS1,δν−ℓ(Ω), 0≤δ<1. This statement is inspired in a new characterization of Hardy-Littlewood-Sobolev inequalities for elliptic and canceling homogeneous operators A(D) with constant coefficients that extends and unifies several results stemming from the classical Hardy-Sobolev estimates. Variants and applications are presented with focus on operators associated to elliptic systems of complex vector fields.

Asymptotic shear and the intrinsic conformal geometry of null-infinity

In this article, we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presence of gravitational waves can be understood to be a generalization of the tractor calculus of conformal manifolds adapted to the case of degenerate conformal metrics. It follows that the whole formalism is, by construction, manifestly conformally invariant. We first show that a choice of asymptotic shear amounts to a choice of linear differential operator of order 2 on the bundle of scales of null-infinity. We refer to these operators as Poincaré operators. We then show that Poincaré operators are in one-to-one correspondence with a particular class of tractor connections, which we call “null-normal” (they generalize the normal tractor connection of conformal geometry). The tractor curvature encodes the presence of gravitational waves, and the non-uniqueness of flat null-normal tractor connections corresponds to the “degeneracy of gravity vacua” that has been extensively discussed in the literature. This work thus brings back the investigation of the radiative phase space of gravity to the study of (Cartan) connections and associated bundles. This should allow us, in particular, to proliferate invariants of the phase space.