On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut

In the numerical solution of the diffraction problem for an acoustic plane wave in a half-plane with a cut, boundary conditions that are equivalent to the radiation conditions at infinity are set in a neighborhood of the points of the cut. Joining the physical boundary conditions on the cut, a closing set of equations of order 4N, where N is the number of grid points on the cut, is obtained. The so-called Green’s grid function for the half-plane is used, which makes it possible to pass from one grid layer to another one for the solution satisfying certain conditions at infinity.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

On some properties of the free boundary in a neighborhood of the points of contact with a given boundary

For the simplest elliptic obstacle problem, the behavior of the free boundary in the vicinity of the points where it meets the prescribed boundary of a domain is studied. The previous result of the author on the C¹ smoothness of the boundary ∂N of the noncoincidence set is improved. The Lipschitz condition on ∂N assumed earlier is shown to be superfluous. Bibliography: 4 titles. Dedicated to O. A. Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 303–312. Translated by I. Kostin.     

Determination of the boundary of the plastic zone in a mineral vein weakened by a three-dimensional hole

We propose a method of computing the boundary of the plastic zone formed in a neighborhood of the hole during the mining of a mineral. The problem is studied in a three-dimensional formulation. The boundary of the plastic zone is determined from the condition of continuity of the vertical normal stresses acting on the surface of contact of an elastic half-space and an elastoplastic layer. The computation is carried out for a hole having a parallelepipedal shape. One figure. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 6–10, 1991.     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

Cooling of a composite slab in a two-fluid medium

A mixed boundary value problem associated with the diffusion equation that involves the physical problem of cooling of an infinite parallel-sided composite slab in a two-fluid medium, is solved completely by using the Wiener-Hopf technique. An analytical solution is derived for the temperature distribution at the quench fronts being created by two different layers of cold fluids having different cooling abilities moving on the upper surface of the slab at constant speedv. Simple expressions are derived for the values of the sputtering temperatures of the slab at the points of contact with the respective layers, assuming the front layer of the fluid to be of finite width and the back layer of infinite extent. The main problem is solved through a three-part Wiener-Hopf problem of a special type and the numerical results under certain special circumstances are obtained and presented in the form of a table.     

Continuity of the free boundary in a non-degenerate p-obstacle problem type with monotone solution

We prove the continuity of the free boundary for a non-degenerate p-obstacle problem with monotone solution. The proof uses techniques of comparison and the growth of the solution near free boundary points.     

Limit theorems for the diameter of a random sample in the unit ball

We prove a limit theorem for the maximum interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit d-dimensional ball for d≥2. The results are specialised for the cases when the points have spherical symmetric distributions, in particular, are uniformly distributed in the whole ball and on its boundary. Among other examples, we also give results for distributions supported by pointed sets, such as a rhombus or a family of segments.      

Nonuniqueness of a Solution to the Problem on Motion of a Rigid Body in a Viscous Incompressible Fluid

This paper is devoted to the problem on motion of a rigid body in a viscous incompressible fluid. It is proved that there exist at least two weak solutions of this problem if collisions of the body with the boundary of the flow domain are allowed. These solutions have different behavior of the body after the collision. Namely, for the first solution, the body goes away from the boundary after the collision. In the second solution, the body and the boundary remain in contact. Bibliography 15 titles.To Vsevolod Alekseevich Solonnikov on the occasion of his jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 199–209.     

Error Estimates for the Boundary Element Approximation of a Contact Problem in Elasticity

The two-dimensional frictionless contact problem of linear isotropic elasticity in the half-space is treated as a boundary variational inequality involving the Poincare–Steklov operator and discretized by linear boundary elements. Quadratic growth of the energy near the solution is shown under weak regularity assumptions if the central axis of external forces intersects the boundary of the domain in a Lebesgue null set. Estimating the numerical range of the discrete Poincare–Steklov operator and properly modifying it, enables the application of an error estimate for semi-coercive variational inequalities. Optimal order of convergence is obtained in the underlying Sobolev space H^1/2(∂Ω)². © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

On the Limits at the Martin Boundary for a Class of Functions

In this paper we prove a criterion for existence of pathwise limits at the Martin boundary for functions with gradient in L _loc ². (This implies that such functions have fine limits at almost all Martin boundary points.)     

Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the gummel map

The initial/boundary-value problem for isothermal, lattice, semiconductor device modeling is described and analyzed. This nonlinear elliptic/parabolic system of reaction/diffusion/convection type is determined by a Maxwell equation, relating space/charge and the electric field, and by two continuity equations for the free electron and hole carrier concentrations. The Einstein relations for Brownian motion are not assumed in this analysis, so that the electrostatic potential, u, and the carrier concentrations, n and p, are the fundamental dependent variables of the system. The boundary conditions are Dirichlet conditions for dependent variable values on the contact portions of the device, and homogeneous Neumann conditions, expressing insulation, on the complement. Complicating the analysis are the transition singularity points between the mixed boundary conditions, and the field dependence of the mobility and diffusion coefficients. By means of a physically motivated analysis of the convective current component, we are able to uncouple the system by a cyclic horizontal line analysis, without an unreasonable time step restriction. The corresponding linear equations are solved by a contractive inner iteration. The outer iteration is shown to converge to a unique solution of the system, under singularity classification at the transition points. The definition of this outer iteration follows the steady-state Gummel iteration at discrete time steps. An existence theory is a by-product of the analysis, and is separated from uniqueness theory.     

On the behavior of free boundaries near the boundary of the domain

Let u be a solution to the obstacle problem in a domain Ω⊂ℝ ⁿ . In this paper, the behavior of the free boundary in a neighborhood of ϖΩ is studied. It is proved that under some conditions the free boundary touches ϖΩ at contact points. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 5–19. Translated by T. N. Rozhkovskaya.     

On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction

In this paper we complement recent work of Maischak and Stephan on adaptive hp-versions of the BEM for unilateral Signorini problems, respectively on FEM-BEM coupling in its h-version for a nonlinear transmission problem modelling Coulomb friction contact. Here we focus on the boundary element method in its p-version to treat a scalar variational inequality of the second kind that models unilateral contact and Coulomb friction in elasticity together. This leads to a nonconforming discretization scheme. In contrast to the work cited above and to a related paper of Guediri on a boundary variational inequality of the second kind modelling friction we take the quadrature error of the friction functional into account of the error analysis. At first without any regularity assumptions, we prove convergence of the BEM Galerkin approximation in the energy norm. Then under mild regularity assumptions, we establish an a priori error estimate that is based on a novel Céa–Falk lemma for abstract variational inequalities of the second kind.     

Homogenization of a fluid problem with a free boundary

The stationary Stokes equations with a free boundary are studied in a perforated domain. The perforation consists of a periodic array of cylinders of size and distance O(ε). The free boundary is given as the graph of a function on a two‐dimensional perforated domain. We derive equations for the two‐scale limit of solutions. The limiting equation is a free boundary system. It involves a nonlinear eliptic operator corresponding to the nonlinear mean‐curvature expression in the original equations. Depending on the equation for the contact angle, the pressure is in general unbounded. © 2000 John Wiley & Sons, Inc.     

Multiple single-peaked solutions of a class of semilinear Neumann problems via the category of the domain boundary

For a smooth domain with compact boundary we investigate the problem with Neumann boundary conditions, where f has superlinear but subcritical growth. Provided that is sufficiently small we show the existence of at least positive solutions with single maximum points that lie on . We replace the standard variational setting used in the case of homogeneous f by considering the restriction of the free functional to a suitable submanifold of the Sobolev Space . Received May 25, 1997 / Accepted October 3, 1997