Quasibounded solutions to the complex Monge-Ampère equation
Mårten Nilsson
TL;DR
The paper develops a robust pluripotential framework for the Dirichlet problem for the complex Monge-Ampère equation with singular boundary data on B-regular domains, introducing pluri-quasibounded boundary data and solving in the Blocki–Cegrell class $D(Ω)$ via envelope and comparison principles. It proves existence and uniqueness of pluri-quasibounded solutions for the inhomogeneous problem with compliant measures and analyzes propagation of boundary singularities into the interior through refined pluripolar hulls, yielding continuity off the propagating hull. In the homogeneous case it characterizes maximal pluri-quasibounded solutions, showing that a b-pluripolar boundary set is necessary and sufficient for existence and continuity results. Together, the results generalize harmonic Poisson representations and provide a comprehensive approach to unbounded boundary data in pluripotential theory with implications for complex geometry on regular domains.
Abstract
We study the Dirichlet problem for the complex Monge-Ampère operator on a B-regular domain $Ω$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $Ω$ and $\partial Ω$, defined by the existence of plurisuperharmonic majorants that dominate their absolute value in a strong sense - that is, the ratio of the function to the majorant tends to zero as the function tends to infinity. For such data, we prove existence and uniqueness of solutions in the Blocki--Cegrell class $\mathcal{D}(Ω)$, using a recently established comparison principle. In the unit disk, our approach recovers harmonic functions represented as Poisson integrals of $L^1$ boundary data with respect to harmonic measure, and our characterization extends to all regular domains in $\mathbb{R}^n$, when the boundary data is continuous almost everywhere. We also describe how boundary singularities propagate into the interior via a refined pluripolar hull.