Bernard Brighi

Approximated convex envelope of a function

The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science.

On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium

In this paper we investigate the similarity solutions of a plane mixed convection boundary layer flow near a semi-vertical plate, with a prescribed power law function of the distance from the leading edge for the temperature, that is embedded in a porous medium. We show the existence and uniqueness of convex and concave solutions for positive values of the power law exponent.

On a Similarity Boundary Layer Equation

The purpose of this paper is to study the autonomous third order nonlinear differential equation f ′′′+ m+1 2 ff ′′−mf ′2 = 0 on (0,∞), subject to the boundary conditions f(0) = a ∈ R, f ′(0) = 1 and f ′(t) → 0 as t → ∞. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as free convection in porous medium or flow adjacent to a stretching wall. Our goal here is to investigate by a direct approach this boundary value problem as completely as possible, say studying existence or non-existence and uniqueness or non-uniqueness of solutions according to the values of the real parameter m. In particular, we will emphasize similarities and differences between the cases a = 0 and a 6= 0 in the boundary condition f(0) = a.

Recent Advances on Similarity Solutions Arising During Free Convection

This paper reviews results about free convection near a vertical flat plate embedded in some saturated porous medium. We focus on a third order autonomous differential equation that gives a special class of solutions called similarity solutions. Two cases are under consideration: in the first one we prescribe the temperature on the plate and in the second one we prescribe the heat flux on it. We will also see that the same equation appears in other industrial processes.

Elliptic and parabolic problems

The goal of these proceedings and of the meeting of Gaeta was to celebrate and honor the mathematical achievements of Haim Brezis. The prodigious influence of his talent and his personality in the domain of nonlinear analysis is unanimously acclaimed! This impact is visible in the huge number of his former students (dozens), students of former students (hundreds) and collaborators (hundreds). Thus the Gaeta meeting was, to some extent, the family reunion of part of this large community sharing a joint interest in the field of elliptic and parabolic equations and pushing it to a very high standard. Italy has a long tradition and taste for analysis and we could not find a better place neither a more complete support for the realisation of our pro ject. We have to thank here the university of Cassino, Napoli, Roma “la Sapienza”, the GNAMPA-Istituto di Alta Matematica, CNR-IAC, MEMOMAT, RTN Fronts-Singularities, the commune of Gaeta. Additional founding came from the universities of Mulhouse and Zurich. Finally, we are grateful to Birkh¨ auser and Dr. Hempfling who allowed us to record the talks of this conference in a prestigious volume.

On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behaviour of the solutions of the autonomous third-order non-linear differential equation f‴+(m+2)f f″−(2m+1)f′2=0 on ℝ+ with the boundary conditions f(0)=−γ, f′(∞)=0 and f″(0)=−1. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing-up co-ordinates to obtain a plane dynamical system. Copyright

Finite differences on triangular grids

We define a finite differences method for triangular grids and we show how to link it to a finite element method. From this new point of view we then analyze properties of the solution and convergence.

STEADY FREE CONVECTION IN A BOUNDED AND SATURATED POROUS MEDIUM

In this paper we are interested with a strongly coupled system of partial differential equations that modelizes free convection in a two-dimensional bounded domain filled with a fluid saturated porous medium. This model is inspired by the one of free convection near a semi-infinite impermeable vertical flat plate embedded in a fluid saturated porous medium. We establish the existence and uniqueness of the solution for small data in some unusual spaces.

Approximation of infima in the calculus of variations

Abstract The goal of this paper is to give numerical estimates for some problems of the Calculus of Variations in the nonhomogeneous scalar case. The stored energy function considered is then a function ϕ : Ω × R n → R . We try to compare the infimum of the energy defined by ϕ on a Sobolev space, with the infimum of the same energy on a finite element space, in terms of the mesh size.

On a Nonlinear System of PDE’s Arising in Free Convection

By employing a fixed point argument, we prove existence and uniqueness of the weak solution to a nonlinear system of PDE’s arising in free convection. An adapted weak formulation is involved to let the solution as weak as possible and avoid any additional smoothness hypotheses.