Continuity of envelopes of unbounded plurisubharmonic functions

Abstract

AbstractOn bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $$\phi $$ ϕ which is continuous in the extended reals on the closure of the domain. For $$\phi $$ ϕ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron–Bremermann envelope $$P \phi $$ P ϕ 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–Ampère equation $$\begin{aligned} (dd^cu)^n = \mu \end{aligned}$$ ( d d c u ) n = μ being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values continuous in the extended reals and unbounded from above.