Boundary singularities of plurisubharmonic functions

We study the Perron–Bremermann envelope P(μ, φ):=sup{u(z) ; u ∈ PSH(Ω), (dd^cu)ⁿ≥ μ, u^* ≤ φ} on a B-regular domain Ω. Such envelopes occupy a central position within pluripotential theory as they, for suitable μ and φ harmonic and continuous on the closure of Ω, constitute unique solutions to the Dirichlet problem for the complex Monge–Ampère operator. Much is also known about the measures that guarantee that the solution is continuous, but the corresponding problems for unbounded or discontinuous φ have received very little attention. This is the main theme of this thesis. 

In paper I and II, by adapting and expanding Leutwiler and Arsove's theory of quasibounded harmonic functions, we introduce a set of positive plurisubharmonic functions which may be approximated from below by functions in L^∞(Ω) ∩ PSH(Ω) outside a pluripolar set. This approximation property is exploited to show that P(μ, φ) is continuous outside a pluripolar set for a large class of measures, given that φ is bounded from below, is continuous in the extended reals, and have a non-trivial strong majorant, i.e. a plurisuperharmonic majorant whose singularities in a precise sense surpass those of φ. We also show that  P(μ, φ) then corresponds to a unique solution to a Dirichlet problem with unbounded boundary data.

In paper III, we show that the Dirichlet problem is uniquely solvable for bounded boundary data with a b-pluripolar discontinuity set, by modifying an extended version of the comparison principle due to Rashkovskii. We also show that the discontinuity set being b-pluripolar is not necessary for the uniqueness. In particular, we construct a class of boundary data for which the Dirichlet problem is uniquely solvable, but where the Lebesgue measure of the set of discontinuities is positive.

In paper IV, we discuss two variations of Edwards' theorem. We prove one version of the theorem for cones not necessarily containing all constant functions, and in particular, we allow the functions in the cone to have a non-empty common zero set. In the other variation, we replace suprema of point evaluations and infima over Jensen measures by suprema of other continuous functionals and infima over a set measures defined through a natural order relation induced by the cone. As applications, we give some results on propagation of discontinuities for Perron–Bremermann envelopes on hyperconvex domains, as well as a characterization of minimal elements in the order relation mentioned above.