On Carathéodory boundary extension of mappings with moduli inequality

The paper establishes a Carathéodory-type boundary extension theorem for open discrete mappings satisfying a Poletsky inverse modulus inequality. Let f : D → D′ be an open discrete mapping between domains in Rn (n ≥ 2) that satisfies the inequality M(Γf(y0, r1, r2)) ≤ ∫ Q(y) · ηn(|y − y0|) dm(y) at every point y0 ∈ D′. Under the assumptions that D has a weakly flat boundary, Q is locally integrable on spheres in a suitable sense, the cluster set C(f, ∂D) lies in a closed set E ⊂ D′ with respect to which D′ is locally finitely connected, and f−1(E ∩ D′) is nowhere dense in D, the mapping f admits a continuous extension f : D → D′ with f(D) = D′. The result generalizes classical boundary extension theorems to a broader class of mappings with controlled distortion and is published in.