On the integrability of the Jacobian under minimal hypotheses

Grand Orlicz spaces and global integrability of the Jacobian

“…Our results (Theorem 3.1 and Theorem 3.5) are analogs to the corresponding local estimates by Müller [15], Iwaniec and Sbordone [12], Greco [8], Iwaniec and Martin [11], Koskela and Zhong [14]. Note also that our technique could be applied to the problem of local integrability, but the results obtained will not be more general than those in Koskela and Zhong [14].…”

“…The grand Lebesgue spaces were introduced by Iwaniec and Sbordone [12] and they found many applications in Analysis, see [12], [10]. The small Lebesgue spaces were introduced by A. Fiorenza [5] as associate to grand spaces.…”

Mappings of finite distortion: Reverse inequalities for the Jacobian

“…Suppose f is a mapping with exponentially integrable distortion. Again the known results (see [24], [25], [16], [17], [20]) tell us that the assumptions of Lemma 5.3 on f are satisfied. Pick p such that n − 1 < p < n and…”

“…Under the assumptions f ∈ W 1,n loc and K ∈ L n−1+ε loc it is known that the mapping f of finite distortion is continuous, discrete and open and satisfies the condition N and that distributional Jacobian of f is equal to the pointwise Jacobian of f (see [10], [18], [28], [24], [25] and [20]). …”

Estimates for the topological degree and related topics

“…We finally mention that the Jacobian determinant was extensively studied in the literature; see, e.g., [4,5,28,38,39,44,51,52,64,65,66,67,80,81,85,86] and the references therein.…”