Complex Dirichlet problem with unbounded boundary values

Abstract

We study the Dirichlet problems with unbounded boundary data. Our main theorem proves interior continuity from a local biholomorphic deformation satisfying a one-sided compatibility condition near the infinite locus and barriers on the associated boundary cluster set. The deformed domains need not be nested, and the proof uses neither a Runge hypothesis nor a pseudoconvex ambient neighbourhood. We also prove nonmonotone Perron--Bremermann convergence outside an intrinsic neighbourhood pluripolar hull, a disk-fibre hull representation, stability under simultaneous nonmonotone variations of the domain and obstacle, and uniqueness in a fixed obstacle class.