Boundary pairs associated with quadratic forms

We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet‐to‐Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non‐smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.     

Stability of a Numerov type finite–difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis

We consider an initial-boundary value problem for the one-dimensional nonstationary Schr?dinger equation on the half-axis and study a two-level symmetric finite-difference scheme of Numerov type with higher approximation order. This scheme is constructed on a finite mesh, which is uniform with respect to space, with a nonlocal approximate transparent boundary condition of a general form (of Dirichlet-to-Neumann type). We obtain assertions about the stability of the finite-difference scheme in two norms with respect to the initial data and free terms in the equation and in the approximate transparent boundary condition under suitable conditions in the form of inequalities on the operator of approximate transparent boundary condition. Bibliography: 12 titles.     

Existence and uniqueness of solutions of hyperbolic equations in divergence form with various boundary conditions on various parts of the boundary

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.     

A trace theorem for the Dirichlet form on the Sierpinski gasket

In connection with the theory for Brownian motion on fractals, a corresponding Dirichlet form has been defined. We consider here the fractal known as the Sierpinski gasket, and characterize the trace of the domain of the Dirichlet form to the boundary of the gasket, boundary in this context meaning the triangle which confines the gasket.     

The Green Function for Elliptic Systems in Two Dimensions

We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. Our main goal is construct the Green function for the operator with mixed boundary conditions in a Lipschitz domain. Thus we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We require a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary conditions. Our proof proceeds by defining a variant of the space BMO(Ω) that is adapted to the boundary conditions and showing that the solution exists in this space. We also give a construction of the Green function with Neumann boundary conditions and the fundamental solution in the plane.     

Singular boundary integral equations of boundary value problems of elastic dynamics in the case of subsonic running loads

We consider boundary value problems for an elastic medium bounded by a cylindrical surface on which a load with a constant-in-time form is moving at a constant subsonic velocity. This class of problems is a model class for the dynamics of underground structures like transport tunnels and ground transport. The method of generalized functions is developed for the solution of boundary value problems. We construct dynamic analogs of Green formulas and use them to derive singular boundary integral equations, which solve the considered boundary value problems.     

Integral representation of a solution of the Neumann problem for the Stokes system

The Neumann problem for the Stokes system is studied on a domain in R ³ with Ljapunov bounded boundary. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series.     

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self‐adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self‐adjoint operators. We then characterize the lower bounded self‐adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.     

On the nonlinear dependence of the optimal boundary control on the initial and terminal conditions

We consider the optimal boundary control of string vibrations and study the case of boundary force control at one end of the string, the other end being free. We show that if the initial and terminal data are arbitrary, then the dependence of the optimal boundary control on these data may be nonlinear. Conditions under which the dependence is linear are indicated. An example is considered in which the optimal control can be found in closed form.     

Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.     

Jordanian deformation of the open XXX spin chain

We find the general solution of the reflection equation associated with the Jordanian deformation of the SL(2)-invariant Yang R-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same K-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.     

Smoothness of solutions of a special one-sided problem

We consider a planar domain, namely a curvilinear quadrilateral. We study a variational inequality of special form on the set of functions that are monotonically increasing on part of the boundary. This problem corresponds to a one-sided problem for an elliptic equation. A boundary condition of first kind is prescribed on part of the boundary, while on the other part of the boundary the tangential derivative is nonnegative and the product of the tangential and oblique derivatives is zero. We establish that the first derivatives of the solution satisfy a Hölder condition. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 173–186.     

Inverse coefficient problems for elliptic equations in a cylinder: I

We study sufficient conditions for the unique solvability of the inverse coefficient problem. We obtain various global sufficient conditions in the form of constraints on the signs of the given functions and their derivatives. As a corollary, we consider statements of inverse coefficient problems with overdetermination on the boundary, where the Dirichlet conditions are supplemented with the vanishing condition for the normal derivative on part of the boundary.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

Analyticity of a free boundary in one problem of axisymmetric flow

We prove the solvability of a boundary-value problem in the case where the Bernoulli condition is given on a free boundary in the form of an inequality. We establish the analyticity of the free boundary. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1692–1700, December, 1998.     

Classical solution of a one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary

We consider a nonlinear parabolic boundary value problem of the Stefan type with one space variable, which generalizes the model of hydride formation under constant conditions. We suggest a grid method for constructing approximations to the unknown boundary and to the concentration distribution. We prove the uniform convergence of the interpolation approximations to a classical solution of the boundary value problem. (The boundary is smooth, and the concentration distribution has the necessary derivatives.) Thus, we prove the theorem on the existence of a solution, and the proof is given in constructive form: the suggested convergent grid method can be used for numerical experiments.     

American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach

We consider the American option pricing problem in the case where the underlying asset follows a jump‐diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro‐partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary.     

ON THE REGULARITY UP TO THE BOUNDARY OF SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS

Abstract

We consider the regularity up to the boundary of solutions of those elliptic boundary value problems that can be treated by the non-variational theory developed by Schechter and others. The usual method of differential quotients is replaced by a method of Schappel [5] which is based on the properties of a Laplace-type operator 1-?2. In this paper the results of Schappel, which are given for boundary value problems in variational form, are extended to the non-variational case. We make important use of fractional order Sobolev spaces on the boundary and trace theorems.     

On the energetic Galerkin boundary element method applied to interior wave propagation problems

We consider two-dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potential. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the related energetic Galerkin boundary element method. Numerical results are presented and discussed.     

Inverse coefficient problems for elliptic equations in a cylinder: II

We study sufficient conditions for the unique solvability of the inverse coefficient problem. We obtain various global sufficient conditions in the form of constraints on the signs of the given functions and their derivatives. As a corollary, we consider statements of inverse coefficient problems with overdetermination on the boundary, where the Dirichlet conditions are supplemented with the vanishing condition for the normal derivative on part of the boundary. We prove sufficient conditions for the existence of a solution in this case.