Weak limit of homeomorphisms in W^1,n−1: Invertibility and lower semicontinuity of energy

¹ Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czech Republic
² Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

^* Corresponding author: anna.a.dolezalova@jyu.fi

Received: 11 September 2023
Accepted: 13 January 2024

Abstract

Let Ω, Ω′ ⊂ ℝⁿ be bounded domains and let f_m: Ω → Ω′ be a sequence of homeomorphisms with positive Jacobians J_fm > 0 a.e. and prescribed Dirichlet boundary data. Let all f_m satisfy the Lusin (N) condition and sup_m ∫_Ω( |D f_m|^{n - 1} + A( |cof D f_m|) + φ(J_f)) < ∞, where A and φ are positive convex functions. Let f be a weak limit of f^m in W^1,n−1. Provided certain growth behaviour of A and φ, we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.