In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problemwhere the d a t a / a n d u 0 belong to L l (Q x (0, T)) and L'(O), and where the function a:(0, T) x Q x R w -> R N is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space i/(0, T; Wj' p (O)) into its dual space. The renormalised solution is an element of C°([0, T]; 1/(0)) such that its truncates T K (u) belong to L"(0, T, Wj' p (O)) with lim \Du\ p dxdt = 0; x~+ ™ JKSMSK + I this solution satisfies the equation formally obtained by using in the equation the test function S(u)
We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis x 3) and we investigate the limit behavior of this problem as the periodicity ε and the radius r of the rods tend to zero (see fig.1 below). We use a decomposition of the displacement field in the rods of the form u = U + u where the principal part U is a field which is piecewise constant with respect to the variables (x 1 , x 2) (and then naturally extended on a fixed domain), while the perturbation u remains defined on the oscillating domain containing the rods. We derive estimates of U and u in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to ε and r, of the constant in Korn's inequality in such an oscillating domain. To deal with the field u, we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate. Résumé Nous considérons un ensemble de poutresélastiques périodiquement distribuées sur une plaqueélastique 3d (toutes d'axe x 3) et nous analysons le comportement limite de ce problème lorsque la périodicité ε et le rayon r des poutres tendent vers zéro. Nous introduisons une décomposition du champ de déplacement de la forme u = U + u dans laquelle la partie principale U est un champ constant par morceau par rapport aux variables (x 1 , x 2) (et qui s'étend donc naturellement sur un domaine fixe), alors que la perturbation u reste un champ défini sur le domaine oscillant qui représente les poutres. Nous donnons des estimations de U et u en fonction de l'énergieélastique totale. Ceci permet d'obtenir des estimations a priori de u sans chercheràévaluer la dépendance, par rapportà ε et r, de la constante de l'inégalité de Korn pour un tel domaine oscillant. Pour traiter le champ u, nous utilisons une version d'opérateur d'éclatement qui permet simultanément de redimensionner toutes les poutres et de travailler sur le
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