Equivalence of generalized and least-maximal-plurisubharmonic boundary conditions

Determine whether the boundary condition that the least maximal plurisubharmonic majorant of a plurisubharmonic function equals the maximal boundary datum, namely \(\widetilde u=\phi\), implies the stronger condition \(\widetilde{u-\phi}\equiv 0\) for functions in the relevant Cegrell classes.

Background

The paper studies a generalized Dirichlet problem for the complex Monge–Ampère equation with possibly unbounded maximal plurisubharmonic boundary data $\phi$. The natural boundary condition is formulated using the least maximal plurisubharmonic majorant: $\widetilde{u-\phi}\equiv 0$.

The authors note that this condition always implies $\widetilde u=\phi$, but the converse is not established. Resolving the converse would clarify whether the apparently weaker requirement that the least maximal plurisubharmonic majorant of $u$ equal $\phi$ is actually equivalent to the boundary condition used in the definition of $\mathcal E^1_\phi(\Omega)$.