On some nonlinear expressions of the Jacobian. (English) Zbl 0983.46034

Greco, Luigi; Verde, Anna

Differ. Integral Equ. 13, No. 10-12, 1569-1582 (2000).

Summary: We define the expressions \(J\log^{1+ \alpha}(e+|Df|)\) and \(J\log^{1+ \alpha}(e+|J|)\) as Schwartz distributions, for \(f: \Omega\subset\mathbb{R}^n\to \mathbb{R}^n\) a Sobolev mapping such that \(|Df|^n\log^\alpha(e+ |Df|)\) is locally integrable, \(-1<\alpha<0\), and \(J\) the Jacobian determinant.

MSC:

46F10	Operations with distributions and generalized functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38	Linear operators on function spaces (general)
46B70	Interpolation between normed linear spaces
30C80	Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

Schwartz distributions; Sobolev mapping; locally integrable; Jacobian determinant