On the Index Formula for an Isometric Diffeomorphism

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah–Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold.     

A Thomson’s Principle and a Rayleigh’s Monotonicity Law for Nonlinear Networks

We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.     

Schwarz-type problem of nonhomogeneous Cauchy-Riemann equation on a triangle

We consider the Schwarz-type boundary-value problem (BVP) of the nonhomogeneous Cauchy-Riemann equation on an isosceles orthogonal triangle. By the technique of plane parqueting and the Cauchy-Pompeiu formula on the triangle, the Schwarz-Poisson formula is obtained. We also investigate boundary behaviors of the Schwarz-type operator and the Pompeiu-type operator. Especially, boundary-values at the corners are proved to exist. Finally, the solution of the Schwarz-type BVP is explicitly obtained.     

A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit N-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin’s results obtained for the two-dimensional case by methods of function theory are transferred to the N-dimensional case (N ≥ 3). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.     

Convolution solutions of an abstract stochastic Cauchy problem

We study convolution solutions of an abstract stochastic Cauchy problem with the generator of a convolution operator semigroup. In the case of additive noise, we prove the existence and uniqueness of a weak convolution solution; this solution is described by a formula generalizing the classical Cauchy formula in which the solution operators of the homogeneous problem are replaced by the convolution solution operators of the homogeneous problem. For the problem with multiplicative noise, we find a condition under which the weak convolution solution coincides with the soft solution and indicate a sufficient condition for the existence and uniqueness of a weak convolution solution; the latter can be obtained by the successive approximation method.     

The Forward–Backward Algorithm and the Normal Problem

The forward–backward splitting technique is a popular method for solving monotone inclusions that have applications in optimization. In this paper, we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward–backward operator. We also provide a formula for the range of the displacement map of the forward–backward operator. Several examples illustrate our theory.     

Growth Estimates for Semigroups Generated by 2Sx2 Operator Matrices

In this paper we give an explicit formula for semigroups generated by 2×2 operator matrices. Using this formula we can derive an estimate for the growth bound of the associated matrix generator. Finally these results are applied to a complete second order Cauchy problem yielding an explicit formula and a growth estimate for the solution of this problem.     

Non-linear eigenvalue problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.     

On variations in the solutions of integro-differential equations of mixed problems of elasticity theory for domain variations

The behaviour of the solution of the boundary value problem for a pseudodifferential equation (PDE), Green's function of this problem, and also some of their local and global characteristics, during variation of the domain is investigated. Formulas are proposed that enable the solution of a broad class of PDE in a domain to be expressed in terms of the solution in the near domain. Local characteristics of the solution are expressed in terms of the local characteristics of the solution in the near domain. A double asymptotic form of Green's function for both arguments tending to the domain boundary occurs in the variation formula. The variation of this double asymptotic form as the domain varies is expressed in terms of this same asymptotic form. The system of variation formulas obtained is closed. It enables the PDE solution in the domain to be reduced to the solution of an ordinary differential equation in functional space. The local characteristics of the solution can also be found by this method without calculating the solution itself. If there is sufficient symmetry in the initial operator, then conservation laws in the Noether sense are obtained for its Green's function and its asymptotic form. The behaviour of the quantities under investigation is studied under inversion.

The investigation of variations of the solutions of problems for the variation of the domain occurs in the paper by Hadamard /1/, who studied the variation in conformal mapping and obtained a formula similar to (1.4). The formula for the variation of the solution of the boundary value problem for an elliptic differential equation is obtained in /2/. Variation formulas for the case of the operator of the problem about a crack and a circular domain are obtained in /3, 4/. The Irwin formula /5/ is obtained from formulas (1.4) and (1.21) by substitution.     

Legendre wavelets approach for numerical solutions of distributed order fractional differential equations

In this study, a new numerical method for the solution of the linear and nonlinear distributed fractional differential equations is introduced. The fractional derivative is described in the Caputo sense. The suggested framework is based upon Legendre wavelets approximations. For the first time, an exact formula for the Riemann–Liouville fractional integral operator for the Legendre wavelets is derived. We then use this formula and the properties of Legendre wavelets to reduce the given problem into a system of algebraic equations. Several illustrative examples are included to observe the validity, effectiveness and accuracy of the present numerical method.     

An Operator Formula for the Shift of the Solution of the Cauchy Problem for an abstract Euler-Poisson-Darboux Equation

Mathematical Notes - We establish an operator formula for the shift of a solution in the case where the summand in the Euler-Poisson-Darboux equation containing the first derivative of the solution...     

Finite-dimensional stabilization of stationary Navier-Stokes systems

We consider the stabilization problem for an unstable solution of an operator equation of Navier-Stokes type. We show that one can exponentially stabilize this solution by treating it as the unique solution of a stationary variational inequality; the stabilizing operator has finite-dimensional range.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

A generalized Green's formula for elliptic problems in domains with edges

The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 106–147.     

m-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.     

A Multisensor Deconvolution Problem

Meisters and Peterson gave an equivalent condition under which the multisensor deconvolution problem has a solution when there are two convolvers, each the characteristic function of an interval. In this article we find additional conditions under which the deconvolution problem for multiple characteristic functions is solvable. We extend the result to the space of Gevrey distributions and prove that every linear operator S, fromthe space of Gevrey functions with compact support onto itself, which commutes with translations can be represented as convolution with a unique Gevrey distribution T of compact support. Finally, we find explicit formula for deconvolvers when the convolvers satisfy weaker conditions than the equivalence conditions using nonperiodic sampling method.     

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

We study a semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrödinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrödinger operator on a path space over a Riemannian manifold.