Weak limit of homeomorphisms in W^1,n-1 and (INV) condition

2021/12/15 by Doležalová, Anna, Hencl, Stanislav, Malý, Jan
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2112.08041

Abstract

Let Ω,Ω'⊂ℝ³ be Lipschitz domains, let f_m:Ω→Ω' be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and sup_m ∫_Ω(|Df_m|²+1/J²_fm)<∞. Let f be a weak limit of f_m in W^1,2. We show that f is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition 1/J²_f∈ L¹ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of W^1,2 Sobolev homeomorphisms in ℝ³ are not the same.

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