Sobolev-like Spaces Research Articles

A two-dimensional boundary value problem of elliptic type with nonlocal conjugation conditions

Abstract We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.

Total variation with nonlocal FT-Laplacian for patch-based inpainting

We consider the problem of inpainting for relatively large damaged areas where the best-known results are achieved using patches. We stem from the ROF-type model. We propose a new ROF model with nonlocal operators and modify it with the F-transform-based operators. As a result, the minimization is considered over a searching space restricted to a finite set of possible reconstructions; each of them is a result of a patch-based inpainting. The fidelity term in the proposed ROF model is estimated by the norm in a Sobolev-like space, which increases the overall quality of reconstruction.

Parabolic-hyperbolic transmission problem in disjoint domains

In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper we investigate a mixed parabolic-hyperbolic initial-boundary value problem in two nonadjacent rectangles with nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problem the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.

New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

ABSTRACTIn this paper, we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper, we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints.

A fast algorithm for the numerical solution of an integral equation with logarithmic kernel

We describe an algorithm for the numerical solution of an integral equation of the form ? 1/? ?1?1 [(y ? x) ln |y ? x| ? h(x, y)] u(y) dy/?1?y2 = f(x), ?1 < x < 1, which is based on a collocation-quadrature method and which has the same convergence rate as this method, but only O(n log n) complexity. This integral equation turns out to be an ill-posed problem in (the best possible choice of) a pair of non-periodic Sobolev-like spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm.

Integer topological charges for finite energy fields in the O(3) σ-model

In the (2+1)-dimensional classical O(3) σ-model, all finite energy fields have integer topological charges regardless of their asymptotic behavior at infinity. The topological charge is conserved for the fields with the finite Euclidean action, without further assumptions on the field equations or asymptotics. The fields with continuous first derivatives as well as fields in Sobolev-like spaces analogical to W1, 2 are considered.

Numerical solution of a two-dimensional hyperbolic transmission problem

An initial boundary value problem for a two-dimensional hyperbolic equation in two disjoint rectangles is investigated. The existence and uniqueness and a priori estimates for weak solutions in appropriate Sobolev-like spaces are proved. Few finite difference schemes approximating this problem are proposed and analyzed.

Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity

We study the limiting behavior of a nonlinear Schrödinger equation describing a 3-dimensional gas that is strongly confined along the vertical, z direction. The confinement induces fast oscillations in time, that need to be averaged out. Since the Hamiltonian in the z direction is merely assumed confining, without any further specification, the associated spectrum is discrete but arbitrary, and the fast oscillations induced by the nonlinear equation entail countably many frequencies that are arbitrarily distributed. For that reason, averaging cannot rely on small denominator estimates or like. To overcome these difficulties, we prove that the fast oscillations are almost-periodic in time, with values in a Sobolev-like space that we completely identify. We then exploit the existence of long-time averages for almost-periodic functions to perform the necessary averaging procedure in our nonlinear problem.

Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension

We study a nonlinear Dirac system in one space dimension with a quadratic nonlinearity which exhibits null structure in the sense of Klainerman. Using an $L^{p}$ variant of the $L^2$ restriction method of Bourgain and Klainerman-Machedon, we prove local well-posedness for initial data in a Sobolev-like space $\hat{H^{s,p}}(\R)$ whose scaling dimension is arbitrarily close to the critical scaling dimension.

Controllability and Optimization of Pseudohyperbolic Systems

The problems of singular optimization and controllability of systems described by the boundary-value problems for pseudohyperbolic equations are studied. The existence and uniqueness theorems for solutions to basic boundary-value problems for pseudohyperolic equations in the nonclassic Sobolev-like spaces under wide assumptions as to right-hand sides are proved. Based on these results, final impulse-point controllability of the systems and existence of optimal control are studied.

A fast algorithm for solving the generalized airfoil equation

The present paper is devoted to the quadrature method for Cauchy-type singular integral equations on [-1, 1] with an additional logarithmic singularity of the kernel. An algorithm being based on this method is given whose realization only requires a computational work of O( n ln n) operations in order to determine n unknown coefficients of the approximate solution. Furthermore, in a scale of weighted Sobolev-like spaces on [-1, 1] (having analogous properties like the classical Sobolev spaces on the unit circle) the order of convergence is optimal.

On a class of partial differential equations with singular coefficients

We study the equation div(fΔu)=0 for functions f which may become infinite at some subset of the domain of definition. After defining adequate Sobolev-like spaces of functions, we make a careful study of the behaviour of admissible solutions in the neighbourhood of singular points. This enables us to show that the Dirichlet problem possesses a space of solutions whose dimension is determined by the number and characteristics of the connected components of the set f=∞, a similar conclusion holds for the Neumann problem.

Polynomial Approximation of Functions in Sobolev Spaces

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

Polynomial approximation of functions in Sobolev spaces

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

Some results on the boundary regularity of harmonic functions

Let θ be an open set in the plane which contains the interval I: {(x, y)∣y=0, -1?x?1}. We consider functions u, which are harmonic in θ-I and continuous in θ. Then, without additional smoothness conditions on f(x)=u(x,0),-1?x$le;1, the one sided normal derivatives(?u/?y)+limh?0+u(x,h)-f(x)/h and (?u/?y)-=limh?0-u(x,h)-f(x)/h, may not exist at any point of I. Here we assume only that f(x) is continuous. We show that in this case the normal derivatives will exist in a Sobolev-like Space of distributions.