September  2007, 8(2): 493-510. doi: 10.3934/dcdsb.2007.8.493

Distributional convergence of null Lagrangians under very mild conditions

1. 

Centre de Mathématiques INSA de Rennes & IRMAR, 20 ave. des Buttes de Coësmes, 35043 Rennes Cedex, France

2. 

Dipartimento di Matematica, Università di Roma, La Sapienza, P.le A. Moro 2, 00185 Rome, Italy

Received  January 2007 Revised  April 2007 Published  June 2007

We consider sequences $U^\epsilon$ in $W^{1,m}(\Omega;\RR^n)$, where $\Omega$ is a bounded connected open subset of $\RR^n$, $2\leq m\leq n$. The classical result of convergence in distribution of any null Lagrangian states, in particular, that if $U^\ep$ converges weakly in $W^{1,m}(\Omega)$ to $U$, then det$(DU^\epsilon)$ converges to det$(DU)$ in $\D'(\Omega)$. We prove convergence in distribution under weaker assumptions. We assume that the gradient of one of the coordinates of $U^\epsilon$ is bounded in the weighted space $L^2(\Omega,A^\epsilon(x)dx;\RR^n)$, where $A_\epsilon$ is a non-equicoercive sequence of symmetric positive definite matrix-valued functions, while the other coordinates are bounded in $W^{1,m}(\Omega)$. Then, any $m$-homogeneous minor of the Jacobian matrix of $U^\epsilon$ converges in distribution to a generalized minor provide that $|A_\epsilon^{-1}|^{n/2}$ converges to a Radon measure which does not load any point of $\Omega$. A counter-example shows that this latter condition cannot be removed. As a by-product we derive improved div-curl results in any dimension $n\geq 2$.
Citation: Marc Briane, Vincenzo Nesi. Distributional convergence of null Lagrangians under very mild conditions. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 493-510. doi: 10.3934/dcdsb.2007.8.493
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