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Plurisubharmonic functions with discontinuous boundary behavior

Mårten Nilsson

TL;DR

The paper addresses solving the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data by linking solvability and uniqueness to the boundary discontinuities being b-pluripolar on B-regular domains; it develops an extended domination principle and Perron–Bremermann envelopes to construct solutions. It proves that uniqueness is equivalent to the boundary set being b-pluripolar, with sharp results in one complex dimension where discontinuities with Lebesgue measure zero align with b-pluripolarity, and it shows that in higher dimensions these conditions can be relaxed, allowing boundary behavior to be prescribed on sets of arbitrarily small measure. The paper then establishes continuity of envelopes on Reinhardt domains, extending Walsh’s theorem, and demonstrates continuity for Perron envelopes with densities in L^p via Kołodziej-type capacity methods. Finally, in the unit ball, it shows that non–b-pluripolar boundary discontinuities can still yield unique, interior-continuous solutions, and constructs boundary data with arbitrarily small boundary measure that fix the interior PSH function, highlighting the limits of a.e. continuity or b-pluripolarity as necessary conditions for uniqueness.

Abstract

We study the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex Monge-Ampère measures. This result is optimal in the unit disk, as boundary functions with b-pluripolar discontinuity then coincides with functions that are continuous almost everywhere. We also show that neither of these properties of the boundary function - being continuous almost everywhere or having discontinuities forming a b-pluripolar set - are necessary conditions in order to establish uniqueness and continuity of the solution in higher dimensions. In particular, there are situations where it is enough to prescribe the boundary behavior at a set of arbitrarily small Lebesgue measure.

Plurisubharmonic functions with discontinuous boundary behavior

TL;DR

The paper addresses solving the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data by linking solvability and uniqueness to the boundary discontinuities being b-pluripolar on B-regular domains; it develops an extended domination principle and Perron–Bremermann envelopes to construct solutions. It proves that uniqueness is equivalent to the boundary set being b-pluripolar, with sharp results in one complex dimension where discontinuities with Lebesgue measure zero align with b-pluripolarity, and it shows that in higher dimensions these conditions can be relaxed, allowing boundary behavior to be prescribed on sets of arbitrarily small measure. The paper then establishes continuity of envelopes on Reinhardt domains, extending Walsh’s theorem, and demonstrates continuity for Perron envelopes with densities in L^p via Kołodziej-type capacity methods. Finally, in the unit ball, it shows that non–b-pluripolar boundary discontinuities can still yield unique, interior-continuous solutions, and constructs boundary data with arbitrarily small boundary measure that fix the interior PSH function, highlighting the limits of a.e. continuity or b-pluripolarity as necessary conditions for uniqueness.

Abstract

We study the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex Monge-Ampère measures. This result is optimal in the unit disk, as boundary functions with b-pluripolar discontinuity then coincides with functions that are continuous almost everywhere. We also show that neither of these properties of the boundary function - being continuous almost everywhere or having discontinuities forming a b-pluripolar set - are necessary conditions in order to establish uniqueness and continuity of the solution in higher dimensions. In particular, there are situations where it is enough to prescribe the boundary behavior at a set of arbitrarily small Lebesgue measure.
Paper Structure (4 sections, 11 theorems, 76 equations)

This paper contains 4 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Boundary values with b-pluripolar discontinuity
  3. Continuity of envelopes on Reinhardt domains
  4. Large discontinuity sets in the unit ball

Key Result

Lemma 2.1

Let $u,v \in \mathcal{PSH}(\Omega)\cap L^\infty(\Omega), \Omega \Subset \mathbb{C}^n$ and suppose that where $F \subset \partial \Omega$ is b-pluripolar. If $(dd^c v)^n \leq (dd^c u)^n$, then $u \leq v$ on $\Omega$. In particular, if $\lim_{z \rightarrow \zeta}(u(z) - v(z)) = 0$ for all $\zeta \in \partial \Omega \setminus F$ and $(dd^c v)^n = (dd^c u)^n$, then $u = v$.

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark
  • Theorem 2.3
  • proof
  • Remark
  • Remark
  • Example 2.4
  • Theorem 3.1
  • ...and 16 more