Smooth dependence on data of solutions and contact regions for a Signorini problem

We prove that the solutions to a 2D Poisson equation with unilateral boundary conditions of Signorini type as well as their contact intervals depend smoothly on the data. The result is based on a certain local equivalence of the unilateral boundary value problem to a smooth abstract equation in a Hilbert space and on an application of the Implicit Function Theorem to that equation.     

Solution branches of a semilinear elliptic problem at corank-2 bifurcation points

Bifurcations of a semilinear elliptic problem on the unit square with the Dirichlet boundary conditions are studied at corank-2 bifurcation points. We show the existence of bifurcating solution branches and their parameterizations via a nonsingular enlarged problem.     

Analysis of a Signorini problem with friction

This paper deals with the study of a nonlinear problem of frictionalcontact between an elastic body and a rigid foundation. Theelastic constitutive law is assumed to be nonlinear and thecontact is modelled with Signorini's conditions and a versionof Coulomb's law of dry friction. We present two weak formulationsof the problem and establish existence and uniqueness results,using arguments of elliptic variational inequalities and a fixed-pintproperty. Moreover, we prove some equivalence results and studythe behaviour of the solution when the coefficient of frictiontends to zero.     

Optimal regularity for the Signorini problem

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C ^1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C ^1,β hypersurface, β > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren’s monotonicity formula and the optimal regularity of global solutions.     

A quasi-static Signorini contact problem for a thermoviscoelastic beam

This paper is concerned with the existence, uniqueness and numerical solution of a system of equations modelling the evolution of a quasi-static thermoviscoelastic beam that may be in contact with two rigid obstacles. A finite element approximation is proposed and analysed and some numerical results are given. Work partially supported by the Brazilian institution CNPq.     

Regularity for the nonlinear Signorini problem

We study the regularity of the solution to a fully nonlinear version of the thin obstacle problem. In particular we prove that the solution is C1,α for some small α>0. This extends a result of Luis Caffarelli of 1979. Our proof relies on new estimates up to the boundary for fully nonlinear equations with Neumann boundary data, developed recently by the authors.     

Existence of a solution for a Signorini contact problem for Maxwell-Norton materials

The aim of this article is to study the quasistatic evolutionof a Maxwell–Norton three-dimensional viscoelastic solidwith contact constraints. After introducing the appropiate functionalframework, we will discretize the problem in time using an implicitscheme whose resultant variational inequality is well posed.By using monotonicity arguments together with compensated compactnesstechniques, we will prove that the corresponding discrete solutionconverges to a solution of the continuous problem.     

Multiple bifurcation branches for variational inequalities

We prove the existence of two bifurcation branches for a variational inequality in a case when the corresponding asymptotic problem is nonsymmetric. We use a nonsmooth variational framework and a blow-up argument which allows to find multiple critical points possibly at the same level. An application to plates with obstacle is presented.     

Semi-smooth Newton methods for the Signorini problem

Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.     

A coupling of FEM-BEM for a kind of Signorini contact problem

In this paper, we consider a kind of coupled nonlinear problem with Signorini contact conditions. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive an asymptotic error estimate of the approximation to the coupled FEM-BEM variational inequality. Then we design an iterative method for solving the coupled system, in which only three standard subproblems without involving any boundary integral equation are solved. It will be shown that the convergence speed of this iteration method is independent of the mesh size.     

The method of fundamental solutions for Signorini problems

We investigate the use of method of fundamental solutions (MFS)for the numerical solution of Signorini boundary value problems.The MFS is an ideal candidate for solving such problems becauseinequality conditions alternating at unknown points of the boundarycan be incorporated naturally into the least-squares minimizationscheme associated with the MFS. To demonstrate its efficiency,we apply the method to two Signorini problems. The first isa groundwater flow problem related to percolation in gentlysloping beaches, and the second is an electropainting application.For both problems, the results are in close agreement with previouslyreported numerical solutions.     

Signorini problem in Hencky plasticity

Sufficient conditions are given, in order to have an equilibrium displacement field for an elasto-plastic body satisfying a constraint at the boundary. The author has been partially supported by a National Research Project of M.P.I. and by I.A.N. of C.N.R.     

Hybrid finite element methods for the Signorini problem

We study three mixed linear finite element methods for the numerical simulation of the two-dimensional Signorini problem. Applying Falk's Lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them. Two of these finite elements provide optimal results under reasonable regularity assumptions on the Signorini solution, and the numerical investigation shows that the third method also provides optimal accuracy.

     

Half-stable solution branches for ordinary bifurcation problems

It is shown that second order bifurcation problems with a positive, autonomous nonlinearity have a smooth branch of positive solutions which tends to infinity. Moreover, this branch satisfies a stability rule saying that the solutions are stable if the branch turns to the right and unstable if it turns to the left.     

Boundary augmented Lagrangian method for the Signorini problem

An augmented Lagrangian method, based on boundary variational formulations and fixed point method, is designed and analyzed for the Signorini problem of the Laplacian. Using the equivalence between Signorini boundary conditions and a fixed-point problem, we develop a new iterative algorithm that formulates the Signorini problem as a sequence of corresponding variational equations with the Steklov-Poincaré operator. Both theoretical results and numerical experiments show that the method presented is efficient.     

Smooth solutions of an initial-value problem for some differential difference equations

The subject matter of this paper is an initial-value problem with an initial function for a linear differential difference equation of neutral type. The problem is to find an initial function such that the solution generated by this function has some given smoothness at the points multiple of the delay. The problem is solved using a method of polynomial quasisolutions, which is based on a representation of the unknown function in the form of a polynomial of some degree. Substituting this into the initial problem yields some incorrectness in the sense of degree of polynomials, which is compensated for by introducing some residual into the equation. For this residual, an exact analytical formula as a measure of disturbance of the initial-value problem is obtained. It is shown that if a polynomial quasisolution of degree N is chosen as an initial function for the initial-value problem in question, the solution generated will have smoothness not lower than N at the abutment points.     

Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk

Let D⊂R² be a disk, and let f∈C³. We assume that there is a∈R such that f(a)=0 and f^′(a)>0. In this article, we are concerned with the Neumann problem