On a Kipriyanov problem for a singular ultrahyperbolic equation

We obtain a differential equation for the spherical means generated by a multidimensional generalized shift of an arbitrary smooth “even” function. We study the Asgeirsson property of solutions of a singular ultrahyperbolic equation that includes singular differential operators Δ _B acting in Euclidean spaces, in general, of distinct dimensions. We represent the structure of a “radial” solution of the considered equation. A theorem similar to the Asgeirsson inverse theorem is proved.     

Lagrange's early contributions to the theory of first-order partial differential equations

In 1776, J. L. Lagrange gave a definition of the concept of a “complete solution” of a first-order partial differential equation. This definition was entirely different from the one given earlier by Euler. One of the sources for Lagrange's reformulation of this concept can be found in his attempt to explain the occurrence of singular solutions of ordinary differential equations. Another source of the new definition is contained in an earlier treatise of Lagrange [1774] in which he elaborated an approach to first-order partial differential equations briefly indicated by Euler. The method of “variation of constants,” which was fundamental to his argument, suggested to Lagrange the reformulation of the concept of a “complete solution.” In the present paper I shall discuss both sources of the new definition of “completeness.”     

Existence of solutions to derivative-dependent,nonlinear singular boundary value problems

This article examines two-point boundary value problems (BVPs) for second-order, singular ordinary differential equations where the right-hand-side of the differential equation may depend on the derivative of the solution. We introduce a method to obtain a priori bounds on all potential solutions, including their “derivatives”, to the singular BVP under consideration. The approach is based on the application of differential inequalities of singular type. The ideas are then applied to yield new existence results for solutions.     

A conic manifold perspective of elliptic operators on graphs

We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second-order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators.     

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.     

Coarse-scale representations and smoothed Wigner transforms

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and/or semiclassical limits of wave equations.We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting “smoothed operators” are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand–Shilov spaces, is necessary.Similarly we treat the “smoothed Wigner calculus”. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g., Hartree and cubic nonlinear Schrödinger), as coarse-scale phase-space equations (e.g., smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.     

The equivalence of viscosity and distributional subsolutions for convex subequations — a strong Bellman principle

There are two useful ways to extend nonlinear partial differential inequalities of second order beyond the class of C ² functions: one uses viscosity theory and the other uses the theory of distributions. This paper considers the convex situation where both extensions can be applied. The main result is that under a natural “second-order completeness” hypothesis, the two sets of extensons are isomorphic, in a sense that is made precise.     

Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete

The stability ofL ²-eigenvalues and associated eigenspaces of singular second order differential operators of Schrödinger-type is shown for asymptotic perturbations of the coefficients and the domain of definition. The perturbations involved are more general than those studied in [3] and [5], because we do not postulate the convergence of the coefficients “from above” or of the domains “from inside” or “from outside”. Moreover, the domain of definition is allowed to be perturbed in its interior. The underlying abstract perturbation theory was established in a previous paper [9].     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Singular soliton operators and indefinite metrics

We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ～ g(g + 1)/x ². Here g is the genus of spectral curve.     

Differential equations for the elementary 3-symmetric Chebyshev polynomials

We continue the study of a “compound model of a generalized oscillator” and related elementary 3-symmetric Chebyshev polynomials. For these polynomials, we obtain second-order differential equations which are of Fuchs type and have 13 singular points. In the considered simplest case, the obtained results give us an answer to a more general question: What changes in the differential equations for polynomials of the Askey–Wilson scheme when the Jacobi matrix related to these polynomials is perturbed by a diagonal matrix with a complex diagonal? Bibliography: 8 titles.     

Extremal Functions on Sturm–Liouville Hypergroups

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev spaces associated with a singular second-order differential operator. Next, we come up with some results regarding the multiplier operators for the Sturm–Liouville transform.     

Type I operators and the approximation of singular two-point boundary value problems

A class of self adjoint operators associated with second order singular ordinary differential expressions, arises naturally when the problem is weakly formulated and integration by parts is performed. We call this class Type I operators. It turns out that this class can be successfully used to tackle numerical approximations of singular two-point boundary value problems. They can also be approximated by regular differential operators in a straightforward manner without having to bring the delicate structure of singular differential operators to the forefront of the investigation.     

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

An MRA approach to surface completion and image inpainting

The objective of this paper is to introduce a multi-resolution approximation (MRA) approach to the study of continuous function extensions with emphasis on surface completion and image inpainting. Along the line of the notion of diffusion maps introduced by Coifman and Lafon with some “heat kernels” as integral kernels of these operators in formulating the diffusion maps, we apply the directional derivatives of the heat kernels with respect to the inner normal vectors (on the boundary of the hole to be filled in) as integral kernels of the “propagation” operators. The extension operators defined by propagations followed by the corresponding sequent diffusion processes provide the MRA continuous function extensions to be discussed in this paper. As a case study, Green's functions of some “anisotropic” differential operators are used as heat kernels, and the corresponding extension operators provide a vehicle to transport the surface or image data, along with some mixed derivatives, from the exterior of the hole to recover the missing data in the hole in an MRA fashion, with the propagated mixed derivative data to provide the surface or image “details” in the hole. An error formula in terms of the heat kernels is formulated, and this formula is applied to give the exact order of approximation for the isotropic setting.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

The regular part of second-order differential sectorial forms with lower-order terms

We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.     

含奇线(奇面)二阶线性偏微分方程的解在奇线(奇面)附近的性质

本文的第一部分研究了含奇线方程的解在奇线附近的性质;引进了“指数”的概念,从而给出了关于这类方程的“奇型郭西问题”的正确提法;并且通过一种特殊的积分-征分方程的研究,证明了这种“奇型郭西问题”的解的存在性,并且给出其近似解法;最后,就一般的情形,给出了方程一般解的表达式,从而说明了在β+β′<0时,郭西问题的多解性。本文的第二部分研究了空间含奇面方程(?)其中 A_σ是任一祇与变元σ=(σ_1…,σ_n)有关的算子,并且关于(15.5)的奇型郭西问题的解可以用关于方程(不合奇面)(?)(15.6)的郭西问题的解表示出来。同样的方法可用来解决空间却普里金方程(17.1)的郭西问题。     

The differential equations for the feynman amplitude of a single-loop graph with four vertices

A system of second-order partial differential equations for the Feynman amplitude of a single-loop graph with four vertices is obtained. It is proved that the symbol of differential operators of this system is singular (in the sense of I. N. Bernshtein) on the Landau manifold of the Feynman amplitude under consideration. The derived system of differential equations is a multidimensional generalization of the system of differential equations for the hypergeometric function of two variables of Appell and Kampé de Fériet.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 113–119, January, 1978.