Invertibility

Abstract

. Let Ω, Ω ′ ⊂ R n be bounded domains and let f m : Ω → Ω ′ be a sequence of homeomorphisms with positive Jacobians J f m > 0 a.e. and prescribed Dirichlet boundary data. Let all f m satisfy the Lusin (N) condition and sup m R Ω ( | Df m | n − 1 + A ( | cof Df 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.