Speaker: Mårten Nilsson, Faculty of engineering, Lund University
Abstract: In this presentation, we study envelopes (point-wise suprema) of families of plurisubharmonic functions, i.e. subharmonic functions whose compositions with biholomorphic mappings are still subharmonic. Such envelopes occupy a central position within pluripotential theory as they for example (under suitable assumptions) constitute the unique solution to the Dirichlet problem for the complex Monge-Ampère operator. Specifically, we study families defined on a bounded domain in C^n, bounded from above by a function continuous in the extended reals. Given some assumptions on the singularities, we establish a set where the envelope is guaranteed to be continuous. As an application of the methods involved, we are able to construct unique solutions to certain complex Monge-Ampère equations where the boundary data is unbounded.
