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.
Citation
Luigi Greco. Tadeusz Iwaniec. Uma Subramanian. "Another approach to biting convergence of Jacobians." Illinois J. Math. 47 (3) 815 - 830, Fall 2003. https://doi.org/10.1215/ijm/1258138195
Information