Fourth-order Differential Operator Research Articles

On the positive invertibility of a differential operator on a graph

In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.

On the anomaly interpretation of amplitudes in self-dual Yang-Mills and gravity

We investigate the integrability anomalies arising in the self-dual sectors of gravity and Yang-Mills theory, focusing on their connection to both the chiral anomaly and the trace anomaly. The anomalies in the self-dual sectors generate the one-loop all-plus amplitudes of gravitons and gluons, and have recently been studied via twistor constructions. On the one hand, we show how they can be interpreted as an anomaly of the chiral U(1) electric-magnetic-type duality in the self-dual sectors. We also note the similarity, for the usual fermionic chiral anomaly, between the 4D setting of self-dual Yang-Mills and the 2D setting of the Schwinger model. On the other hand, the anomalies in the self-dual theories also resemble the trace anomaly, sharing the same type of non-local effective action. We highlight the role of a Weyl-covariant fourth-order differential operator familiar from the trace anomaly literature, which (i) explains the conformal properties of the one-loop amplitudes, and (ii) indicates how this story may be extended to non-trivial spacetime backgrounds, e.g. with a cosmological constant. Moving beyond the self-dual sectors, and focusing on the gravity case, we comment on an intriguing connection to the two-loop ultraviolet divergence of pure gravity, whereby cancelling the anomaly at one-loop eliminates the two-loop divergence for the simplest helicity amplitudes.

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

A unified weighted inequality for fourth-order partial differential operators and applications

In this paper, we establish a fundamental inequality for fourth order partial differential operator P=△α∂s+β∂ss+Δ2 (▪) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator P with some suitable boundary conditions in the case of β<0 or α≠0,β=0. As application, we obtain a resolvent estimate for P, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.

Inverse problems for the beam vibration equation with involution

This article considers inverse problems for a fourth-order hyperbolic equation with involution. The existence and uniqueness of a solution of the studied inverse problems is established by the method of separation of variables. To apply the method of separation of variables, we prove the Riesz basis property of the eigenfunctions for a fourth-order differential operator with involution in the space ${{L}_{2}}(-1,1)$. For proving theorems on the existence and uniqueness of a solution, we widely use the Bessel inequality for the coefficients of expansions into a Fourier series in the space ${{L}_{2}}(-1,1)$. A significant dependence of the existence of a solution on the equation coefficient $\alpha$ is shown. In each of the cases $\alpha &lt;-1$, $\alpha &gt;1$, $-1&lt;\alpha&lt;1$ representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are written out.

Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.

Asymptotics of the eigenvalues of a fourth order differential operator with eigenvalue dependent and periodic boundary conditions

Asymptotics of the eigenvalues of a fourth order differential operator with eigenvalue dependent and periodic boundary conditions

On the Spectral Properties of a Fourth-Order Self-Adjoint Operator

We consider a spectral problem for a fourth-order differential operator with real periodic coefficients and Neumann-type boundary conditions. For this operator, the eigenvalue asymptotics and a regularized trace formula are obtained.

Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions

In this article we consider the fourth-order operators \(A_1u:=(au'')''\) and \(A_2u:=au''''\) in divergence and non divergence form, where \(a:[0,1]\to\mathbb{R}_+\) degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on \(a\), we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems.

Spectral asymptotics and a trace formula for a fourth-order differential operator corresponding to thin film equation

Spectral asymptotics and a trace formula for a fourth-order differential operator corresponding to thin film equation

Asymptotic Expressions of Fourth Order Sturm-Liouville Operator with Conjugate Conditions

In this paper, it is studied the asymptotic expression of fourth order differential operator with periodic boundary conditions. For this operator, it is also considered conjugate boundary conditions at x=0 which shows discontinuity. For this purpose, firstly asymptotic expression of solutions areobtained. Then by using the the asymptotic formulas of fundamental solutions, asymptotic expression of eigenvalues and eigenfunctions are presented. It is also dealt with the asymptotic expression of same operator with antiperiodic boundary conditions and conjugate conditions

Sharp eigenvalue asymptotics of fourth-order differential operators

Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are obtained.

Modified Morley method by free formula for fourth order elliptic singular perturbation problems

A new modified Morley method is proposed to solve fourth order elliptic singular perturbation problems of two or three dimension. The method has the fewest number of DOFs on each element. Different from the usual finite element methods, we slightly modify the bilinear term corresponding to the second order differential operator and keep that corresponding to the fourth order differential operator and the right-hand side term unchanged. The modification is based on the idea of free formula introduced by Bergan and Nygård in 1984. Theory and numerical results show that the solution is uniformly convergent with respect to the perturbed parameter ε . From the view of computation, our method is efficient since the fewest number of DOFs are employed and the local stiffness matrix takes less computation time than the usual finite element method due to the special construction of free formula.

Oscillatory and spectral properties of fourth-order differential operator and weighted differential inequality with boundary conditions

In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality’s weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero.

Spectral Estimates for the Fourth-Order Differential Operator with Periodic Coefficients

Spectral Estimates for the Fourth-Order Differential Operator with Periodic Coefficients

Dependence of Eigenvalues of Discontinuous Fourth-order Differential Operators with Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a fourth-order differential operator with eigenparameter dependent boundary conditions and transmission conditions. To study the eigenvalues of the problem, we establish a new operator associated with the considered problem. Furthermore, we prove that the eigenvalues are differentiable depending on the parameters of the problem. Finally, the differential expressions of the eigenvalues with respect to all parameters are given.

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

&lt;abstract&gt;&lt;p&gt;This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.&lt;/p&gt;&lt;/abstract&gt;

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Weighted Second-Order Differential Inequality on Set of Compactly Supported Functions and Its Applications

In the paper, we establish the oscillatory and spectral properties of a class of fourth-order differential operators in dependence on integral behavior of its coefficients at zero and infinity. In order to obtain these results, we investigate a certain weighted second-order differential inequality of independent interest.

Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.