Jean-Michel Rakotoson

A compactness Lemma for quasilinear problems: Application to parabolic equations

In this paper, we solve the problem ut + Au + F(u, ▽u) = μ and u(0) = u0 where μ and u0 are bounded Radon measures. The method is based upon three fundamental lemmas: A compactness result (Lemma 3), a regularity result (Lemma 7), and an integration by parts (Lemma 2).

Mathematical treatment of the magnetic confinement in a current carrying stellarator

The model studied concerns the case of a stellarator machine and the magnetic confinement is modeled by using averaging methods and suitable vacuum coordinates. This is shown to lead to a two-dimensional Grad-Shafranov type problem for the averaged poloidal flux function. Various problems are considered and it is pointed out that corresponding problems for models based on tokamak machines are essentially similar.

An optimal compactness theorem and application to elliptic-parabolic systems

Abstract We give a new optimal compactness criterion which insures that time dependent bounded sequences in suitable Hilbert spaces contain convergent subsequences. Our proof is related to PDE techniques. We then give an abstract application of the result.

Petits espaces de Lebesgue et quelques applications

Resume On se propose detablir quelques proprietes des petits espaces de Lebesgue introduits par Fiorenza [7], notamment la convergence monotone de Levi et des proprietes dequivalence de normes. En combinant ces proprietes avec les inegalites de Poincare–Sobolev pour le rearrangement relatif [11], nous donnons quelques estimations precises concernant les espaces de Sobolev associes a ces espaces et les regularites des solutions dequations quasilineaires lorsque les donnees sont dans ces espaces. Pour citer cet article : A. Fiorenza, J.-M. Rakotoson, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 23–26

The Cahn-Hilliard equation for an isotropic deformable continuum

Abstract In this note, we study a Cahn-Hilliard equation in a deformable elastic isotropic continuum. We define boundary conditions associated with the problem and obtain the existence and uniqueness of solutions. We also study the long time behavior of the system.

Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure data

Abstract We extend the results of [1–5] on the uniqueness of solutions of parabolic equations. Our results give also some regularity results which complete the existence results made in [6–8].

Nonlinear parabolic equations with p-growth and unbounded data*

Abstract The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u t - div( a ( t , x , u , D u )) = H ( t , x , u , D u ) - div( g ( t , x )) in Q T =]0, T [×Ω, Ω ⊂ R N , with an initial condition u (0) = u 0 , where u 0 is not bounded, | H ( t , x , u , ξ)⩽ β|ξ| p + f ( t , x ) + βe λ 1 |u| f , |g| p/(p-1) ∈ L r (Q T ) for some r = r {N) ⩾ 1, and - div( a ( t , x , u , D u )) is the usual Leray-Lions operator.

On some Bernoulli free boundary type problems for general elliptic operators

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti-Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.

Nonlocal generalized models for a confined plasma in a Tokamak

Abstract The following model appears in plasma physics for a Tokamak configuration: −Δu + g(u) = 0, u ∈ V = H 0 1 (Ω) ⊗ R , ∫ aΩ au an = I > 0 , where I is a given positive constant, which is equivalent to find a fixed point u = F(u −g(u)) + ϕ0 where F is a compact operator on L 2 (Ω) . According to Grad and Shafranov the nonlinearity g can depend on u ∗ which is the generalized inverse of the distribution function m(t) = measx : u(x) > t = −vb u > t -vb (see [1]). But in these cases the map u → g(u) cannot be continuous on all the space v but only on a nonlinear nonclosed set v 0 . This implies that the standard direct method for fixed point cannot be applied to solve the preceding problem. Nevertheless, using the Galerkin method and a topological argument, we prove that there exists a solution u fixed point of u = F(u − g(u)) + ϕ0 under suitable assumptions on g. The model we treat covers a large new class of nonlinearities including relative rearrangment and monotone rearrangment. The resolution of the concrete model needs an extension of the strong continuity result of the relative rearrangement map made in [2] (see Theorem 1.1 below for the definition and result).

Nonlocal elliptic-parabolic equation arising in the transient regime of a magnetically confined plasma in a Stellarator

Abstract We prove the existence an the regularity of solutions of an elliptic-parabolic equation involving the notions of relative rearrangement and monotone rearrangement. These equations were obtained from 3D MHD systems, taking (in particular) into account the Ohm and Faradays laws and the averaging arguments of Hender and Carreras for obtaining a 2D evolution model. Due to the presence of a strong nonlinearity, we introduce a new notion of solution derived from a new property of the relative rearrangement. This allows us to improve the results obtained in the stationary case by cancelling the condition of smallness of the parameter λ in the pressure.