L¹_loc-convergence of Jacobians of Sobolev homeomorphisms via area formula

2025/01/19 by Zofia Grochulska, Grochulska, Zofia
Mathematics · #28A75 (Primary) 26B15 (Secondary) #Advanced Differential Equations and Dynamical Systems #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2501.11106

openalex publication_date 2025/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that given a sequence of homeomorphisms f_k: Ω→ ℝⁿ convergent in W^1,p(Ω, ℝⁿ), p ≥ 1 for n =2 and p > n-1 for n ≥ 3, to a homeomorphism f which maps sets of measure zero onto sets of measure zero, Jacobians Jf_k converge to Jf in L¹_loc(Ω). We prove it via Federer's area formula and investigation of when |f_k(E)| → |f(E)| as k → ∞ for Borel subsets E \Subset Ω.

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