A study of Jacobians in Hardyd–Orlicz spaces

Abstract

We study the Jacobian determinants J = det(9/*/dxj) of mappings / : n C R 71 -> I " in a Sobolev-Orlicz space W 1 ' i> (Q,R n ). Their natural generalizations are the wedge products of differential forms. These products turn out to be in the Hardy-Orlicz spaces T-i^{O). Other nonlinear quantities involving the Jacobian, such as Jlog \J\, are also studied. In general, the Jacobians may change sign and in this sense our results generalize the existing ones concerning positive Jacobians.

On the Product of Functions in BMO and H^\text{1}

“…First, in PDEs we find various nonlinear differential expressions identified by the theory of compensated compactness, see the seminal work of F. Murat [49] and L. Tartar [59], and the subsequent developments [16,17,25]. New and unexpected phenomena concerning higher integrability of the Jacobian determinants and other null Lagrangians have been discovered [46,29,33,28,21] , and used in the geometric function theory [27,26,1], calculus of variations [58,30], and some areas of applied mathematics, [47,48,62]. Recently a viable theory of existence and improved regularity for solutions of PDEs where the uniform ellipticity is lost, has been built out of the distributional div-curl products and null Lagrangians [25,31] .…”

“…The well-developed study of the Jacobian determinants is concerned with the grand Hardy space H 1) (Ω), see [33,30,32]. Let Ω ⊂ R n be a bounded domain.…”

Weakly differentiable mappings between manifolds

“…Unexpectedly, the uniform bound (1.19) is lost. If the Jacobian changes sign then it still belongs to the Hardy space H 1 (X), a well known result by R. Coifman, P. Lions, Y. Meyer and S. Semmes [9] in R n , see also [33], [36]. Again, in manifold setting the arguments establishing H 1 -regularity of the Jacobian will be more subtle than in R n .…”

Gauge Theory of Faddeev–skyrme Functionals

“…The distributional convergence of wedge products is a well-studied subject and we now recall a relevant result from [39] (see also [29] for a different approach).…”