# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2007, 8 (2) : 493-510. doi: 10.3934/dcdsb.2007.8.493
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