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Plurisubharmonic functions with discontinuous boundary behavior

Nilsson, Mårten LU (2023) In Indiana University Mathematics Journal
Abstract
We study the Dirichlet problem for the complex Monge–Amp`ere
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–Amp`ere measures. This result is optimal in the unit disk, as boundary
functions with b-pluripolar discontinuity then coincide 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... (More)
We study the Dirichlet problem for the complex Monge–Amp`ere
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–Amp`ere measures. This result is optimal in the unit disk, as boundary
functions with b-pluripolar discontinuity then coincide 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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
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Contribution to journal
publication status
in press
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in
Indiana University Mathematics Journal
publisher
Indiana University
ISSN
0022-2518
language
English
LU publication?
yes
id
ea21f4f0-9468-4353-bc64-fa04c501ad2a
alternative location
https://www.iumj.indiana.edu/IUMJ/Preprints/60165.pdf
date added to LUP
2024-02-28 11:04:01
date last changed
2024-06-05 14:09:26
@article{ea21f4f0-9468-4353-bc64-fa04c501ad2a,
  abstract     = {{We study the Dirichlet problem for the complex Monge–Amp`ere<br/>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<br/>Monge–Amp`ere measures. This result is optimal in the unit disk, as boundary<br/>functions with b-pluripolar discontinuity then coincide with functions that are<br/>continuous almost everywhere. We also show that neither of these properties<br/>of the boundary function – being continuous almost everywhere or having discontinuities forming a b-pluripolar set – are necessary conditions in order to<br/>establish uniqueness and continuity of the solution in higher dimensions. In<br/>particular, there are situations where it is enough to prescribe the boundary<br/>behavior at a set of arbitrarily small Lebesgue measure.}},
  author       = {{Nilsson, Mårten}},
  issn         = {{0022-2518}},
  language     = {{eng}},
  publisher    = {{Indiana University}},
  series       = {{Indiana University Mathematics Journal}},
  title        = {{Plurisubharmonic functions with discontinuous boundary behavior}},
  url          = {{https://www.iumj.indiana.edu/IUMJ/Preprints/60165.pdf}},
  year         = {{2023}},
}