Another approach to biting convergence of Jacobians

Abstract

We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class ${\mathscr W}^{1,n}(\Omega,\mathbb{R}^n)$. The novelty of our approach is in using ${\mathscr W}^{1,p}$-estimates with the exponents $1\leqslant p \lt n$, as developed in [IS1], [IL], [I1]. These rather strong estimates compensate for the lack of equi-integrability. The remaining arguments are fairly elementary. In particular, we are able to dispense with both the Chacon biting lemma and the Dunford-Pettis criterion for weak convergence in ${\mathscr L}^1(\Omega)$. We extend the result to the so-called Grand Sobolev setting.

Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ${\mathscr L}^{\frac n{n-1}}(\mathbb{R}^n)$ is also obtained. Possible generalizations to the wedge products of differential forms are discussed.