Continuity of envelopes of unbounded plurisubharmonic functions

Abstract

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $\phi$ such that $\phi^*=\phi_*$ on the closure of the domain. For $\phi$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P \phi$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Amp\`ere equation \[ (dd^cu)^n = \mu \] being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values unbounded from above.