Jacobian of weak limits of Sobolev homeomorphisms

Abstract

Abstract:Let Ω be a domain in ℝ n , where n = , . Suppose that a sequence of Sobolev homeomorphisms f k : Ω → ℝ n with positive Jacobian determinants, J(x, f k ) > , converges weakly in W ,p (Ω, ℝ n ), for some p ⩾ , to a mapping f . We show that J(x, f) ⩾ a.e. in Ω. Generalizations to higher dimensions are also given.

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

“…For further information, refer to [15,17,19]. Weak limits of Sobolev homeomorphisms have received significant attention in recent years, with various studies conducted, including [1,3,6,7,8,9,13,16].…”

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

“…The class of weak limits of Sobolev homeomorphisms was recently characterized in the planar case by Iwaniec and Onninen [18,19] and De Philippis and Pratelli [9]. Moreover, one can study the orientation of mappings in this class [15] or even investigate planar BV weak limits and characterize their set of cavities and fractures [6]. In [24] Molchanova and Vodopyanov studied invertibility a.e.…”