Order Linear Differential Operator Research Articles

Univariate right fractional polynomial high order monotone approximation

Abstract Let f ∈ Cr ([−1,1]), r ≥ 0 and let L * be a linear right fractional differential operator such that L *(f) ≥ 0 throughout [−1,0]. We can find a sequence of polynomials Qn of degree ≤ n such that L *(Qn ) ≥ 0 over [−1,0], furthermore f is approximated right fractionally and simultaneously by Qn on [−1,1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f(r) .

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate urn:x-wiley:0025584X:media:mana201500017:mana201500017-math-0004holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that urn:x-wiley:0025584X:media:mana201500017:mana201500017-math-0009Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.

Linear Differential Operators on Contact Manifolds

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.

Differential operators on the weighted densities on the supercircle S1|1

Over the (1, 1)-dimensional real supercircle, we consider the K(1)-modules Dλ,μk of linear differential operators of order k acting on the superspaces of weighted densities, where K(1) is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules. This work is the simplest superization of a result by Gargoubi and Ovsienko.

Weighted a priori bounds for elliptic operators of Cordes type

We prove some weighted $W^{2,2}$ -a priori bounds for a class of elliptic second order linear differential operators of Cordes type on unbounded domains of $\mathbb {R}^{n}$ , $n\geq2$ .

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated with second order linear degenerate elliptic partial differential operators X with rough coefficients, of the form X=−div(P∇)+HR+S′G+F, where the n×n matrix function P=P(x) is nonnegative definite and allowed to degenerate, R, S are families of subunit vector fields, G, H are vector valued functions and F is a scalar function. We operate in a geometric homogeneous space setting and we assume the validity of certain Sobolev and Poincaré inequalities related to a symmetric nonnegative definite matrix of weights Q=Q(x) that is comparable to P; we do not assume that the underlying measure is doubling. We give a maximum principle for weak solutions of Xu≤0, and we follow this with a result describing a relationship between compact projection of the degenerate Sobolev space QH1,p, related to the matrix of weights Q, into Lq and a Poincaré inequality with gain adapted to Q.

A remark on the stability of $$C^\infty $$ C ∞ -hypoellipticity under lower-order perturbations

We study the problem of perturbations of $$C^\infty $$ -hypoelliptic operators by lower order terms. We show that hypoellipticity with a finite loss of derivatives of a linear partial differential operator $$P$$ (with no assumption on the characteristic set), along with its formal adjoint $$P^*$$ , is stable under perturbations by lower order linear partial differential operators whose order depends on the loss of derivatives.

Analysis as a source of geometry: a non-geometric representation of the Dirac equation**Supported by EPSRC grant EP/M000079/1

Consider a formally self-adjoint first order linear differential operator acting on pairs (two-columns) of complex-valued scalar fields over a four-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful four-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object—an abstract formally self-adjoint first order linear differential operator acting on pairs of complex-valued scalar fields.

UNIVARIATE LEFT FRACTIONAL POLYNOMIAL HIGH ORDER MONOTONE APPROXIMATION

Abstract. Let f ∈ C r ([−1,1]), r ≥ 0 and let L ∗ be a linear left frac-tional differential operator such that L ∗ (f) ≥ 0 throughout [0,1]. We canfind a sequence of polynomials Q n of degree ≤ nsuch that L ∗ (Q n ) ≥ 0over [0,1], furthermore f is approximated left fractionally and simulta-neously by Q n on [−1,1].The degree of these restricted approximationsis given via inequalities using a higher order modulus of smoothness forf (r) . 1. IntroductionThe topic of monotone approximation started in [6] has become a majortrend in approximation theory. A typical problem in this subject is: given apositive integer k, approximate a given function whose kth derivative is ≥ 0 bypolynomials having this property.In [3] the authors replaced the kth derivative with a linear differential oper-ator of order k. We mention this motivating result.Theorem 1. Let h,k,p be integers, 0 ≤ h ≤ k ≤ p and let f be a real function,f (p) continuous in [−1,1] with modulus of continuity ω 1 f (p) ,xthere. Leta j (x), j = h,h + 1,...,k be real functions, defined and bounded on [−1,1]and assume a

Scaling functions in the square Ising model

We show and give the linear differential operators of order , for the integrals which appear in the two-point correlation scaling function of Ising model . The integrals are given in expansion around in the basis of the formal solutions of with transcendental combination coefficients. We find that the expression is a solution of the Painlevé VI equation in the scaling limit. Combinations of the (analytic at ) solutions of sum to . We show that the expression is the scaling limit of the correlation function and . The differential Galois groups of the factors occurring in the operators are given.

Characterization of self-adjoint domains for differential operators with interior singular points

We characterize the self-adjoint domains of general even order linear ordinary differential operators which have finite interior singular points in terms of real-parameter solutions of the differential equation. For the purpose we constructed a direct sum space. By the theory of direct sum space and the decomposition of the corresponding maximal domain, we give this complete and analytic characterization in terms of limit-circle solutions. This is for endpoints which are regular or singular and for arbitrary deficiency index.

A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

Собственная функция оператора Лапласа в +1-мерном симплексе

In order to find eigenfunction of the Laplace operator in regular n+1-dimensional simplex the barycentric coordinates are used. For obtaining this result we need some formulas of the analytical geometry. A similar result was obtained in the earlier papers of the author in a tetrahedron from R 3 and in gipertetrahedron from R 4. Let П be unlimited cylinder in the space R n, its cross-section with hyperplane has a special form. Let L be a second order linear differential operator in divergence form, which is uniformly elliptic and η is its ellipticity constant. Let u be a solution of the mixed boundary value problem in Π with homogeneous Dirichlet and Neumann data on the boundary of the cylinder. In some cases the eigenfunction of the Laplace operator allows us to continue this solution from the cylinder Π to the whole space R n with the same ellipticity constant. The obtained result allows us to get a number of various theorems on the solution growth for mixed boundary value problem for linear differential uniformly elliptical equation of the second order, given in unlimited cylinder with special cross-section. In addition we consider n-1-dimensional hill tetrahedron and the eigenfunction for an elliptic operator with constant coefficients in it.

Solvability and stationarity for the optimal control of variational inequalities with point evaluations in the objective functional

AbstractMotivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non‐convex and non‐differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C‐stationarity in the associated Sobolev space setting. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

We prove weak and strong maximum principles, including a Hopf lemma, for C 2 subsolutions to equations defined by linear, second-order, linear, elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C 2 subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C 2 solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W 1, 2 solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.

Spectral parameter power series for perturbed Bessel equations

A spectral parameter power series (SPPS) representation for regular solutions of singular Bessel type Sturm–Liouville equations with complex coefficients is obtained as well as an SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. It is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the Sturm–Liouville problem. Based on the SPPS representation a new mapping property of the transmutation operator for the considered perturbed Bessel operator is obtained, and a new numerical method for solving corresponding spectral problems is developed. The range of applicability of the method includes complex coefficients, complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator. On a set of known test problems we show that the developed numerical method based on the SPPS representation is highly competitive in comparison to the best available solvers such as SLEIGN2, MATSLISE and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the SPPS method delivers excellent numerical results.

Hyers–Ulam stability of the linear differential operator with nonconstant coefficients

We obtain some stability results for the linear differential operator of order one in Banach spaces. As a consequence we derive a result on Hyers–Ulam stability for the linear differential operator of higher order with nonconstant coefficients.

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let Ω be a domain in ℝ N and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup \((e^{-t A_{V}})_{t\geqslant0}\) on L 2(Ω,dx), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p -contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.

On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

Two-point Taylor approximations of the solutions of two-dimensional boundary value problems

We consider two-dimensional elliptic boundary value problems of the form L(U)=F in Ω⊂R2,Ω bounded and open, with a Dirichlet boundary condition U|∂Ω=H, where L is a second order linear differential operator whose coefficients, as well as the functions F and H are differentiable up to a certain degree. In a recent paper [C. Kesan, Taylor polynomial solutions of second order linear partial differential equations, Appl. Math. Comput. 152 (2004) 29–41], a matrix algorithm is introduced to compute the standard Taylor polynomial of the solution U at a certain point in Ω. We propose an alternative formulation of the problem based on a redefinition of the unknown U and the use of the standard Frobenius method that simplifies the computation of the Taylor coefficients of U. We also consider the use of a two-point Taylor representation of the solution, instead of the classical one-point Taylor representation, which gives a more uniform approximation of the solution.