Calculus of Variations and Geometric Measure Theory

I. Fonseca - G. Leoni - J. MalĂ˝

Weak continuity and lower semicontinuity results for determinants

created on 04 Sep 2003
modified by leoni on 27 Jul 2011


Published Paper

Inserted: 4 sep 2003
Last Updated: 27 jul 2011

Journal: Arch. Ration. Mech. Anal.
Volume: 178
Number: 3
Pages: 411-448
Year: 2005


Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if $\{ u_{n}\} $ is bounded in $W^{1,N-1}\left( A;{R}^{N}\right) $, $\left\{adj D u_{n}\right\} $ is in $ L^{{N}/{N-1}} \left(A;R^{{N}\times{N}}\right)$, $u$ is in $BV\left( A;{R}^{N}\right) $, $u_{n}$ converges strongly to $u$ in $L^{1}\left( A;{R}^{N}\right) $ and $ \det D u_{n}$ converges weakly star in the sense of measures to some measure $m$, then for ${L}^{N}$ a.e. $x$ in $ A$ \[ \det D u\left( x\right) =\left({dm}/{d{L}^{N}}\right)\left( x\right) . \] The result is sharp and counterexamples are provided in the cases where regularity of $\left\{ u_{n}\right\} $ or the type of weak convergence are weakened.