On equicontinuity of mappings with branching defined in family of domains

The paper investigates equicontinuity of sequences of open discrete closed mappings with branching defined on a family of domains. The authors consider the family RQ,δ,p,E(D0, D) of mappings satisfying a ring Q-condition with respect to p-modulus (p ∈ (n−1, n]). Under the assumptions that the sequence of domains Dm is regular with respect to the kernel D0, each Dm is locally connected on its boundary, the family of images is equi-uniform with respect to p-modulus, and Q satisfies either finite mean oscillation in D0 or a divergence integral condition involving its spherical averages, the sequence fm is shown to be uniformly equicontinuous in D. Moreover, there exists a subsequence that converges locally uniformly in D0 to a mapping f that admits a continuous boundary extension. When p = n and an additional uniform separation condition on the chordal diameter of boundary components holds, the limit mapping is boundary-preserving. The result provides new distortion estimates for mappings of families of domains and is published in [2].