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Équations elliptiques non linéaires monotones avec un deuxième membre L 1 ou mesure

François Murat — 1998

Journées équations aux dérivées partielles

On considère le problème : - div a ( x , D u ) = f dans Ω , u = 0 sur Ω , Ω est un ouvert borné de 𝐑 N , où a ( x , ξ ) est une fonction de Carathéodory, monotone en ξ , coercive, qui définit un opérateur dans W 0 1 , p ( Ω ) (avec 1 < p N ), et où f appartient à L 1 ( Ω ) ou est une mesure bornée sur Ω . On introduit une nouvelle définition de la solution de ce problème, la notion de solution renormalisée (ou entropique), et on montre l’existence d’une telle solution et sa continuité par rapport à f . Quand f appartient à L 1 ( Ω ) , on montre en outre que cette solution...

A chain rule formula for the composition of a vector-valued function by a piecewise smooth function

François MuratCristina Trombetti — 2003

Bollettino dell'Unione Matematica Italiana

We state and prove a chain rule formula for the composition T u of a vector-valued function u W 1 , r Ω ; R M by a globally Lipschitz-continuous, piecewise C 1 function T . We also prove that the map u T u is continuous from W 1 , r Ω ; R M into W 1 , r Ω for the strong topologies of these spaces.

A generalized strange term in Signorini’s type problems

Carlos ConcaFrançois MuratClaudia Timofte — 2003

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The limit behavior of the solutions of Signorini’s type-like problems in periodically perforated domains with period ε is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε 0 . In the critical case, it is shown that Signorini’s problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from the...

Finite element discretization of Darcy's equations with pressure dependent porosity

Vivette GiraultFrançois MuratAbner Salgado — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose...

Monotone operators in divergence form with x -dependent multivalued graphs

Gilles FrancfortFrançois MuratLuc Tartar — 2004

Bollettino dell'Unione Matematica Italiana

We prove the existence of solutions to - div a x , grad u = f , together with appropriate boundary conditions, whenever a x , e is a maximal monotone graph in e , for every fixed x . We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a 45 rotation, for every fixed x ; in other words, the graph d a x , e is defined through d - e = φ x , d + e , where φ is a Carathéodory contraction in R N . This definition is shown to be equivalent to the fact that a ( x , ) is pointwise monotone...

A Generalized Strange Term in Signorini's Type Problems

Carlos ConcaFrançois MuratClaudia Timofte — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The limit behavior of the solutions of Signorini's type-like problems in periodically perforated domains with period is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from...

Junction of elastic plates and beams

Antonio GaudielloRégis MonneauJacqueline MossinoFrançois MuratAli Sili — 2007

ESAIM: Control, Optimisation and Calculus of Variations

We consider the linearized elasticity system in a multidomain of 𝐑 3 . This multidomain is the union of a horizontal plate with fixed cross section and small thickness , and of a vertical beam with fixed height and small cross section of radius r ε . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When and r ε tend to zero simultaneously, with r ε ε 2 , we identify the limit problem. This limit problem involves six junction conditions.

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