On Carathéodory theorem for Orlicz-Sobolev classes

The paper proves a Carathéodory-type theorem on prime ends boundary extension for open discrete mappings belonging to Orlicz–Sobolev classes. Let f : D → D′ be an open discrete mapping in W^{1,φ}_loc(D) with f(D) = D′ between bounded domains in R^n (n ≥ 3). Under the assumptions that D is regular, the cluster set C(f, ∂D) lies in a closed set E* ⊂ D′, the preimage E = f^{-1}(E*) is closed and nowhere dense in D, D is finitely connected on E ∪ ∂D and on the prime ends, all components of D′ \ E* have strongly accessible boundaries with respect to α-modulus (n−1 < α ≤ n), the function φ satisfies the Calderón condition ∫1^∞ (t/φ(t))^{1/(n−2)} dt < ∞, the inner dilatation satisfies K{I,α}(x, f) ≤ Q(x) a.e., and Q obeys a divergence integral condition on spheres near the boundary, the mapping f admits a continuous extension f : D_P → D′ (where D_P is the prime ends completion of D) such that f(D_P) = D′. The result extends classical boundary extension theorems to unclosed Orlicz–Sobolev classes and is published in [2].