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Dirichlet's problem in Pluripotential Theory
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we focus on Dirichlet's problem for the complex Monge-Ampère equation. That is, for a given non-negative Radon measure µ we are interested in the conditions under which there exists a plurisubharmonic function u such that (ddcu)n=µ, where (ddc)n is the complex Monge-Ampère operator. If this function u exists, then can it be chosen with given boundary values? Is this solution uniquely determined within a given class of functions?

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik , 2008. , 19 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 40
Keyword [en]
Complex Monge-Ampère operator, currents, Dirichlet problem, pluripotential theory, plurisubharmonic function, subextension
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-1562ISBN: 978-91-7264-443-4 (print)OAI: oai:DiVA.org:umu-1562DiVA: diva2:141392
Public defence
2008-03-19, MA121, MIT, 901 87, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2009-06-17Bibliographically approved
List of papers
1. A comparison principle for the complex Monge-Ampère operator in Cegrell's classes and applications
Open this publication in new window or tab >>A comparison principle for the complex Monge-Ampère operator in Cegrell's classes and applications
2009 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, 5539-5554 p.Article in journal (Refereed) Published
Abstract [en]

In this article we will first prove a result about the convergence in capacity. Next we will obtain a general decomposition theorem for complex Monge-Ampère measures which will be used to prove a comparison principle for the complex Monge-Ampère operator.

Keyword
Complex Monge-Amp\`ere operator, plurisubharmonic function
Identifiers
urn:nbn:se:umu:diva-2981 (URN)10.1090/S0002-9947-09-04730-8 (DOI)
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2017-12-14Bibliographically approved
2. Concerning the energy class εp for 0 < p < 1
Open this publication in new window or tab >>Concerning the energy class εp for 0 < p < 1
2007 (English)In: Annales Polonici Mathematici, ISSN 0066-2216, Vol. 91, no 2-3, 119-130 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-2982 (URN)
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2009-06-17Bibliographically approved
3. Monge-Ampère measures on pluripolar sets
Open this publication in new window or tab >>Monge-Ampère measures on pluripolar sets
2009 (English)In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 92, no 6, 613-627 p.Article in journal (Refereed) Published
Abstract [en]

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kołodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.

Keyword
Complex Monge–Ampère operator, Dirichlet problem, Pluripolar set, Plurisubharmonic function
Identifiers
urn:nbn:se:umu:diva-2983 (URN)10.1016/j.matpur.2009.06.001 (DOI)
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2017-12-14Bibliographically approved
4. Boundary values of functions in Cegrell's class Eψ
Open this publication in new window or tab >>Boundary values of functions in Cegrell's class Eψ
2007 In: Annales Polonici Mathematici, ISSN 0066-2216, Vol. 92, no 1, 69-74 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-2984 (URN)
Available from: 2008-02-26 Created: 2008-02-26Bibliographically approved
5. Pluripolar sets and subextension in Cegrell's classes
Open this publication in new window or tab >>Pluripolar sets and subextension in Cegrell's classes
2008 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 53, no 7, 675-684 p.Article in journal (Refereed) Published
Abstract [en]

In this article, we prove that if E is a complete pluripolar set in Ω, then E = { = −∞} for some  (Ω). Moreover, we study the subextension in Cegrell's class p .

Place, publisher, year, edition, pages
Taylor & Francis, 2008
Keyword
complex Monge-Ampére operator, plurisubharmonic function
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-2985 (URN)10.1080/17476930801966893 (DOI)
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2017-12-14Bibliographically approved
6. On the convergence in capacity on compact Kähler manifolds and its applications
Open this publication in new window or tab >>On the convergence in capacity on compact Kähler manifolds and its applications
2008 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 6, 2007-2018 p.Article in journal (Refereed) Published
Abstract [en]

The main aim of the present note is to study the convergence inCX,ω on a compact Kahler mainfold X. The obtained results are used to studyglobal extremal functions and describe the ω-pluripolar hull of an ω-pluripolarsubset in X.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2008
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-2986 (URN)
Available from: 2008-02-26 Created: 2008-02-26 Last updated: 2017-12-14Bibliographically approved

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