Diffeomorphic approximation of planar Sobolev homeomorphisms in Orlicz-Sobolev spaces

Let Ω⊆R2 be a domain, let Φ be a Δ2 Young function and let f∈W1,Φ(Ω,R2) be a homeomorphism between Ω and f(Ω). Then there exists a sequence of diffeomorphisms fk converging to f in the Sobolev-Orlicz space W1,Φ(Ω,R2). Further for an injective continuous map φ∈W1,Φ(∂(−1,1)2,R2) we find a diffeomorphism in W1,Φ((−1,1)2,R2) that equals φ on the boundary.