Accepted Paper
Inserted: 18 dec 2002
Last Updated: 23 jul 2003
Year: 2002
Abstract:
We give new proof of the theorem of K. Zhang on biting convergence of Jacobian determinants for mapping of Sobolev class. The novelty of our approach is in using some estimates with the exponents $1\leqslant p<n$, as developed in \cite{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 Chacon biting lemma and Dunford-Pettis criterion for weak convergence in $L^1(\Omega)$. We extend the result to the so-called grand Sobolev setting.\par Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in $L^{\frac n{n-1}}(\R^n)$ is also obtained. Possible generalizations to the wedge products of differential forms are discussed.
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