TY - JOUR

T1 - Another approach to biting convergence of Jacobians

AU - Greco, Luigi

AU - Iwaniec, Tadeusz

AU - Subramanian, Uma

PY - 2003

Y1 - 2003

N2 - We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class script W sign 1,n(Ω, ℝn). The novelty of our approach is in using script W sign1,p-estimates with the exponents 1 ≤ p < 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 ℒ 1(Ω). We extend the result to the so-called Grand Sobolev setting. Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ℒn/n-1 (ℝn) is also obtained. Possible generalizations to the wedge products of differential forms are discussed.

AB - We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class script W sign 1,n(Ω, ℝn). The novelty of our approach is in using script W sign1,p-estimates with the exponents 1 ≤ p < 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 ℒ 1(Ω). We extend the result to the so-called Grand Sobolev setting. Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ℒn/n-1 (ℝn) is also obtained. Possible generalizations to the wedge products of differential forms are discussed.

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U2 - 10.1215/ijm/1258138195

DO - 10.1215/ijm/1258138195

M3 - Article

AN - SCOPUS:0346331482

SN - 0019-2082

VL - 47

SP - 815

EP - 830

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

IS - 3

ER -