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Publication 03-CNA-15

Weak Continuity and Lower Semicontinuity Results for Determinants

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
giovanni@andrew.cmu.edu

and

Jan Maly
Department of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovska 83,18675 Praha 8
Czech Republic

Abstract: 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 $ \left\{ u_{n}\right\} $ is bounded in $ W^{1,N-1}\left( \Omega;\mathbb{R}^{N}\right) $, $ \left\{ \mathrm{adj}\nabla u_{n}\right\} \subset L^{\frac{N}{N-1}}\left( \Omega;\mathbb{R}^{{N}\times{N}}\right) ,$ and if $ u\in BV\left( \Omega;\mathbb{R}^{N}\right) $ are such that $ u_{n}\rightarrow u$ in $ L^{1}\left( \Omega;\mathbb{R}^{N}\right) $ and

$\displaystyle \det\nabla u_{n}\overset{\ast}{\rightharpoonup}\mu $

in the sense of measures, then for $ \mathcal{L}^{N}$ a.e. $ x\in\Omega$

$\displaystyle \det\nabla u\left( x\right) =\frac{d\mu}{d\mathcal{L}^{N}}\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.

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The original publication will be available at Springer Link

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