Some Landau–Kolmogorov Type Inequalities for Differential Operators Generated by Polynomials

2021 ◽  
Vol 58 (2) ◽  
pp. 246-262
Author(s):  
Vu Nhat Huy ◽  
Nguyen Ngoc Huy ◽  
Chu Van Tiep
Keyword(s):  
Differential Operator ◽  
Explicit Form ◽  
Differential Operators ◽  
Kolmogorov Type

In this paper, we establish some Landau–Kolmogorov type inequalities for differential operators generated by polynomials in the following formfor all , where 0 < g ≤ p ≤ ∞, and the differential operator P (D) is obtained from the polynomial P (x) by substituting . Moreover, the explicit form of and are given.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green&rsquo;s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

First Order Operators on Manifolds With a Group Action

1996 ◽  
Vol 48 (4) ◽  
pp. 758-776 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Differential Operator ◽  
Group Action ◽  
Lie Group ◽  
Differential Operators ◽  
Order Differential Operator ◽  
Compact Lie Group ◽  
First Order ◽  
Rank 2 ◽  
Homogeneous Operators

AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

Semibounded Extensions of Singular Ordinary Differential Operators

1988 ◽  
Vol 31 (4) ◽  
pp. 432-438
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Boundary Conditions ◽  
Differential Operators ◽  
The Self ◽  
Limit Circle ◽  
Singular Differential Operator ◽  
Ordinary Differential Operators

AbstractThe self-adjoint extensions of the singular differential operator Ly = [(py’)’ + qy]/w, where p < 0, w > 0, q ≧ mw, are characterized under limit-circle conditions. It is shown that as long as the coefficients of certain boundary conditions define points which lie between two lines, the extension they help define has the same lower bound.

scholarly journals A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

1997 ◽  
Vol 145 ◽  
pp. 125-142
Author(s):  
Takeshi Mandai
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Negative Integer ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Real Analytic

Consider a partial differential operator(1.1) where K is a non-negative integer and aj,a are real-analytic in a neighborhood of (0, 0)

scholarly journals The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

2017 ◽  
Vol 24 (1) ◽  
pp. 113-134 ◽  
Author(s):  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Differential Operators ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Mean Curvature Equations

AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

On the existence of nodal domains for elliptic differential operators

1983 ◽  
Vol 94 (3-4) ◽  
pp. 287-299 ◽  
Author(s):  
E. Müller-Pfeiffer
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Quadratic Form ◽  
Unbounded Domain ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
Dirichlet Boundary Conditions ◽  
Friedrichs Extension ◽  
Nodal Domains ◽  
Elliptic Differential Operators

SynopsisWe prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

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