Nonlinear variational formulation for a cavitation problem in lubrication

The fluid behaviour in a journal bearing where cavitation takes place is studied. This leads to solving a nonlinear variational inequality with normal discontinuity of derivative on a part of the free boundary. Under some hypotheses, using penalization, the proof of the existence of a solution is given which is compared with one of a simpler model governed by a classical variational inequality.     

A mixed variational formulation of a contact problem with adhesion

A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.     

A variational formulation for the torsional problem of piezoelastic beams

In this paper a variational formulation is presented for the torsional deformation of homogeneous, linear piezoelectric monoclinic beams. All results of the paper are based on a generalization of the Saint-Venant’s theory of uniform torsion of elastic beams to piezoelastic beams. Variational formulation uses the torsional and electric potential functions as the independent quantities of the considered variational functional. The mechanical meaning of the variational functional defined is also given. Examples illustrate the application of the presented variational formulation. Considered examples are the torsional problem of thin-walled piezoelastic beams with closed cross-section, and the torsion of hollow circular cylinders made of orthotropic piezoelectric material.     

A variational inequality formulation of the environmental pollution control problem

In this paper we develop the time-dependent pollution control problem in which different countries aim to determine the optimal investment allocation in environmental projects and the tolerable pollutant emissions, so as to maximize their welfare. We provide the equilibrium conditions governing the model and derive the evolutionary variational inequality formulation. The existence of solutions is investigated and a numerical example is also presented.     

Extension of Hilbert's tenth problem to some algebraic number fields

We extend the solution of Hilbert's tenth problem to algebraic number fields having one pair of complex conjugated embeddings. The proof is based on the extended method of J. Denef used for totally real algebraic number fields.     

Hadamard's variational formula for a mixed problem and an application to a problem related to a signorini-like variational inequality

We give a variational formula for the solution of a Mixed Boundary Value Problem when the geometrical domain is perturbed (Hadamard's Variational Formula) ; we show that this formula contains singularities ; the result is used to construct a new approach to the optimal control problem for a problem related to a Signorini-like Elliptic Variational Inequality.     

A vorticity-velocity variational formulation for the exterior stokes problem in weighted sobolev spaces

In this article, we present a mixed variational formulation for the exterior Stokes problem in terms of the velocity and vorticity variables in the two and three-dimensional cases. The existence and the uniqueness of solutions of the variational problem are proved, as well as the equivalence with the primitive variables formulation.     

Dual variational formulation of the electrostatic problem in an inhomogeneous anisotropic dielectric

The use of developed prospective dielectric materials in various modern electrotechnical and electrophysical devices requires reliable forecast of attainability of the required level of final characteristics depending on properties of those materials. Such forecast is based (among others) on solution of electrostatic problems in an inhomogeneous anisotropic medium allowing one to estimate the ability to satisfy the qualifying standards for effective characteristics of created materials. A dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric is used to solve these problems.     

On a variational problem with non-differentiable constraints

We study the regularity of minimizers and critical points of the Dirichlet energy under an integral constraint given by a non-differentiable function. We obtain existence of a Lipschitz continuous minimizer for a relaxed problem. In two dimensions, some regularity can also be proved for critical points.     

A variational approach to a shape design problem for the wave equation

The problem of determining the optimal damping set for the stabilization of the wave equation may be not well-posed. By means of a vector variational reformulation and use of gradient Young measures, we present a general methodology to relax this kind of problems. From the optimal Young measure associated with the relaxed problem, we obtain information concerning minimizing sequences for the original problem as well as continuity properties of the relaxed cost function. To cite this article: A. Münch et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On a variational problem with lack of compactness related to the Strichartz inequality

We study a variational problem from nonlinear fiber optics which strongly lacks compactness, due to the absence of a priori bounds in spaces different from . A method is established how to restore this missing compactness by means of the dispersive properties inherent in the problem.Received: 30 October 2002, Accepted: 15 April 2003, Published online: 1 July 2003     

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

A stream-function–vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces

In this paper we derive a mixed variational formulation for the exterior Stokes problem in terms of the vorticity and stream function, or the vector potential in three dimensions. The main steps are the construction of the stream function (or vector potential) and the proof of the Babu?ka–Brezzi ‘inf-sup’ condition. The two- and three-dimensional cases are treated separately because the structure of the stream function differs substantially according to the number of dimensions considered. The conclusion of this work is that if the problem is set in the weighted Sobolev spaces of Hanouzet and Giroire, the analysis of the exterior Stokes problem is quite the same as if the domain were bounded.     

On a variational problem for radial solutions to extremal elliptic equations

On the ball |x| ≤ 1 of R ^m , m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved [see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli [Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.      

Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ?-periodically located subsets G _ε belonging to the boundary ?Ω of the domain Ω ? ?³. We construct a limit (homogenized) problem and prove the strong (in H ₁(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ? → 0 in the so-called critical case.     

New formulation for the high multiplicity asymmetric traveling salesman problem with application to the Chesapeake problem

This paper presents a new model for a special type of traveling salesman problem called the High Multiplicity Asymmetric Traveling Salesman Problem (HMATSP). The formulation adopts a flow-based subtour elimination structure and establishes its validity for this problem. Also, we present computational results to demonstrate the efficacy of our modeling approach. The model is then incorporated as a substructure in a formulation for the lot-sizing problem involving parallel machines and sequence-dependent setup costs, also known as the Chesapeake Problem, and related test problems are solved to optimality for the first time in the literature.