tóm tắt tiếng anh sự liên tục của nghiệm đối với bài toán dirichlet cho toán tử monge ampère phức

MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATION PHAM THI LIEU CONTINUITY OF SOLUTION TO THE DIRICHLET PROBLEM FOR THE COMPLEX MONGE-AMPÈRE OPERATOR Subject:

MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATION

PHAM THI LIEU

CONTINUITY OF SOLUTION TO THE DIRICHLET PROBLEM

FOR THE COMPLEX MONGE-AMPÈRE OPERATOR

Subject: Mathematical Analysis

Code: 9.46.01.02

SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

Ha Noi – 2024

THE THESIS WAS COMPLETED AT:

HA NOI NATIONAL UNIVERSITY OF EDUCATION

Supervisors: Assoc Prof Dr Nguyen Xuan Hong

Referee 1: Prof Dr Sc Le Mau Hai

Hanoi National University of Education

Referee 2: Assoc Prof Dr Ngo Quoc Anh

University of Science, Vietnam National University, Hanoi

Referee 3: Assoc Prof Dr Do Duc Thuan

Hanoi University of Science and Technology

This research has been performed at the Ha Noi National University of Education Date of Oral Presentation:……,… /…… /2024

Copy of this dissertation is available at: - The National Library of Viet Nam - Library of Ha Noi National University of Education

1 Rationale

prob-lems of pluripotential theory It has been studied by many authors from the 1970sto the present Among them, E Bedford, B.A Taylor, Z Blocki, U Cegrell,N.N Cuong, S Kolodziej, P.H Hiep, J Wiegerinck, are typical mathematiciansin this research direction In recent years, besides the research of the complex

operator is one of the important and central problems of pluripotential theory, hasbeen studied in many different cases

In 2014, El Kadiri and Wiegerinck [45] introduced and studied the definition

function (F plurisubharmonic functions) defined on a plurifinely open subset (F

strictly pseudoconvex domain with the simple measure is stydied in papers [3], [4],[20], [36], [39], The authors showed the existence of solution and continuoussolution in this paper However, this Dirichlet problem can be further studied inmore general domain classes

of plurisubharmonic functions in strictly pseudoconvex domain with the generalmeasure has been In 2014, Dinew, Guedj and Zeriahi posed the question of the

17 in [17]) This question was partially solved by Cuong [13] in 2018 and fullycompleted in 2020 (see [14]) The author showed the Dirichlet problem has aglobal solution if it has a sub-solution This problem is continually studied inmany different approaches.

In 1979, Bedford and Taylor [3] showed The Dirichlet problem for the complex

domain with the measure 0 has a unique solution and the solution is the ronBremermann This is the concept that is interested in research interest in manydifferent cases The first result in this direction in the strictly pseudoconvex do-main is given by Bremermann [6] and Walsh [54] Later, Simioniuc v Tomassini[51]studied the PerronBremermann in the unbounded strictly pseudoconvex domain.Recently, Nilsson and Wikstrm [50] have studied the PerronBremermann in thebounded B-regular domain

Per-From the above reasons, we chose the research topic: Continuity of solution

2 Objectives

In the thesis, We reseach the following issue:

smallest topology that makes all plurisubharmonic functions on Ω continuous Westudy the existence of solutions and the continuty of solutions of the Dirichlet

oper-ator in strictly pseudoconvex domains through the existence of local solutions

envelope of plurisubharmonic functions in bounded hyperconvex domains

3 Research Subjects

hypercon-vex domains and F -plurisubharmonic functions

5 Scientific and practical significance of the thesis

The pluripotential theory and plurifinely pluripotential theory are researchdirections that are interested by many authors for their applications in Com-plex Analysis and Differential Geometry, Partial Differential Equation, HyperbolicAnalysis, Complex Dynamics, The thesis researches The Dirichlet problem for

continuity of the PerronBremermann envelope of plurisubharmonic functions inbounded hyperconvex domains The results of the thesis have scientific signif-

icance and contribute to the development of the pluripotential theory and rifinely pluripotential theory as well as the techniques in this theory.

plu-6 Thesis Structure

Structure of thesis consists the parts: Introduction, Overview, Chapters, clusion, List of papers used in thesis, Reference The thesis’ main content consiststhree chapters as follows

Con-Chapter 1 The existence of a solution to The Dirichlet problem for the complex

pseudo-convex domain from the Euclide topology to the plurifinely topology Next, we

Furthermore, we showed the solution isn’t continuous in the usual sense This is a

but not strictly pseudoconvex domain

We prove the existence of a global solution to the Dirichlet problem on a strictlypseudoconvex domain if and only if the problem has a local solution

Chapter 3 Continuity of the PerronBremermann envelope of plurisubharmonicfunctions

In this chapter, the first part we recall some concepts in the Cegrell’s class

-subextensible, A function is said to be strong majorant, A function is said tostrong minorant and some of their properties From there, we prove continuity

functions in B-regular and bounded pseudoconvex domains

In 2014, El Kadiri and Wiegerinck [45] introduced and studied the definition

plurisub-harmonic functions on Ω continuous In this chapter, We consider the Dirichlet

operator is the problem of finding a function u satisfying:

5



continuous When Ω is non-smooth pseudoconvex domains, Hong and Thuy [36]

MA(Ω, f, φ) :=





F - lim

plurisubhar-monic functions in strictly pseudoconvex domain has studied in papers: [3], [4],[20], [39], [36] In those papers, the authors showed the existence of a continu-

continuous on ∂Ω

domains

For the convenience of the reader, we repeat the definition of F -plurisubharmonicfunctions as extension of the plurisubharmonic functions given by El Kadiri [42]in 2003

[−∞, +∞) is said to be F -plurisubharmonic function if u is F -upper

of the subset F -open l ∩ Ω of l is either F -subharmonic or ≡ −∞

The set of all F -plurisubharmonic functions (negative F -plurisubharmonic

Remark 1.1.2 i) Every plurisubharmonic functions defined in an Euclide openset is F -plurisubharmonic functions

In this chapter, we use the following results for the F -plurisubharmonic tions

func-Theorem 1.1.3 [48, func-Theorem 3.1]Let f be a bounded F -plurisubharmonic function in a bounded F -open subset Ω

Euclide containing V In particular f is F -continuous on Ω

F - lim

Then, the function

ω :=





is a F -plurisubharmonic function on Ω

-regular domain is a generalization of the definition of strictly pseudoconvex domainfrom Euclide topology to plurifinely topology

if there is a F -defining function of Ω, i.e a bounded, F -continuous function ϕ on

How-ever, the opposite is not true

domian, we will use some auxiliary results

If the measure µ being of the type

F -PSH(Ω) be finite with w ≤ −v on Ω Then, for every measure µ on the QB(Ω)

on Ω such that

Before presenting the main result, we will introduce the following lemma

such that:

Using Lemma 1.2.3, We show the existence of solutions and the continuty of

MA(Ω, f, φ) is solvable

When Ω is a strictly pseudoconvex domain, the solution of the problem MA(Ω, f, φ)is continuous in the usual sense, this has been proven by Theorem 1.2.1 In this

domains Furthermore, we showed the solution isn’t continuous in the usual sense

Theorem 1.3.1 There exists a BF-regular domain Ω in C such that:

(2) The solutions of the problem MA(Ω, f, φ) can not be continuous in the usual

with p > 1 such that

Chapter 2

the Dirichlet problem for the complex

In this chapter, we consider The Dirichlet problem for the complex

µ be a measure in Ω The Dirichlet problem is the problem of finding a functionu satisfying: We study the problem of finding a plurisubharmonic function u in Ωthat satisfies the following conditions:

M(Ω, µ) :=





lim

this direction tells us that the problem M(Ω, µ) is solvable if Ω is strictly

domains have been widely accepted as the standard domain in which we can

is further studied in [4], where the authors showed the existence of a continuous

11

solution if f is continuous Later, some other authors also generalized the resultabove There is a very important result due to Guedj, Ko lodziej and Zeriahi that

Recently, several other authors have used the technique of [20] to study the

[2], [11]) In the case of manifolds, the problem is studied by [16], [23] and someother authors When µ is arbitrary, the problem becomes much more complicated.This problem remained open up until recently In 2014, Dinew, Guedj and Zeriahi

problem (see Question 17 in [17]) This question was partially solved by [13] in2018 and fully completed in 2020 (see [14])

The main purpose of this paper is to study the conditions under which theproblem can be solved To study this, we will find the properties of measures thatadmit local sub-solutions Our first main result is the following theorem

First, we prove that the Dirichlet problem in a bounded domain has a global solution if the problem has a local sub-solution Next, we will prove the problemM(Ω, µ) when Ω is a strictly pseudoconvex domain with a global solution if theproblem has a local solution

Definition 2.1.1 A function is called sub-solution to M(Ω, µ) if it is a solutionto M(Ω, ν) with some measure ν ≥ µ

We need the following result for a class of smooth increasing convex functions

such that

lim

continuous function u on Ω such that it is plurisubharmonic in Ω and satisfies

The existence of continuous solutions of the problem M(Ω, µ) in the case whereΩ is a strictly pseudoconvex domain has been solved by mathematicians in papers[3] , [4], Until recently in [14], Cuong proved that the Dirichlet problem for the

solution if the problem has a sub-solution

and let µ be a non-negative Borel measure in Ω If the problem M(Ω, µ) has asub-solution then the problem M(Ω, µ) is solvable

Our second main result is the following theorem on the locality of the problemM(Ω, µ)

non-negative Borel measure in Ω Then, the problem M(Ω, µ) is local, i.e., it is

solvable on Ω if only if for every z ∈ Ω, there exists rz > 0 such that it is solvableon Ω ∩B(z, rz).

Chapter 3

Continuity of the Perron-Bremermannenvelope of plurisubharmonic functions

φ : K → [−∞, +∞] be an upper semi-continuous function Perron-Bremermannenvelope is defined by





When Ω is unbounded strictly pseudoconvex, Simioniuc and Tomassini [51] the

is continuous outside a pluripolar set Recall that a function φ : Ω → [−∞, +∞]

φ∗(z) := lim inf

as-sociated PerronBremermann envelope to plurisubharmonic functions on boundedhyperconvex domains Let Ω be a B-regular domain or strictly pseudoconvexdomain and φ is definied

φ :=



In this section, we will give the conditions for the functions u and v of class

a pluripolar set Firstly, we will recall the notation of the Cegrells classes ofplurisubharmonic functions from [7]

continuous, real-valued function on ∂Ω can be expanded to a the plurisubharmonicfunction on Ω and continuous on Ω

continuous function in K can be uniformly approximated in K by continuous

plurisubharmonic functions in a neighborhood of K.(ii) A locally closed subset K is called locally B-regular if for every z ∈ K thereexists a sphere U centered on z such that K ∩ U is the B-regular domain

con-tinuous plurisubharmonic function exhaustion ρ : G → (−∞, 0), that is, thecontinuous plurisubharmonic function ρ satisfies {z ∈ G, ρ(z) < c} is a relativelycompact subset of G for each c ∈ (−∞, 0)

The following theorem of Sibony gives us the relationship of B-regular domainin Definition 3.1.1 and Definition 3.1.2

and ∂Ω be a compact B-regular then Ω be a B-regular domain On the contrary,if Ω is a B-regular domain then it is a hyperconvex domain, and if we add the

relation-ship:

A strictly pseudoconvex domain → A strictly hyperconvex domain → A convex domain → A pseudoconvex domain

Denote by F (Ω) the family of plurisubharmonic functions u in Ω such that there

satisfies

sup

j≥1Z

Ω

(ddcuj)n < +∞

Furthermore, let Fa(Ω) denote those functions u ∈ F (Ω) for which

Z

K

(ddcu)n = 0

for all pluripolar sets K ⊂ Ω

of φ on Ω

-subextensible

u1 ∈ Fsea(Ω1) and u2 ∈ Fsea(Ω2), the function

φ :=





belongs to Fsea(Ω1 ∪ Ω2).Proposition 3.1.11 together with Proposition 3.1.10 gives

φ :=





belongs to Fsea(Ω1 ∪ Ω2).Now, we recall the following definition from [49]

if

A function γ is said to be strong majorant to φ, if −γ is strong minorant to −φ

Remark 3.1.14 If φ is bounded from below on Ω then every plurisubharmonicfunction ψ in Ω is strong minorant to φ

Theorem 3.1.15 [49, Theorem 4.8] Let Ω be a bounded B-cregular domain andlet φ e a function having a strong minorant v and a strong majorant w such that

Proposition 3.1.16 Let Ω be a bounded subset of Cn Then, for every φ ∈

φ :=





We are now able to give the proof continuity of the PerronBremermann envelopeof plurisubharmonic functions in bounded hyperconvex domain

on Ω

φ :=



semi-continuous function Assume that there exists a plurisubharmonic function ψ inΩ such that ψ ≤ φ in Ω and

is plurisubharmonic in Ω and satisfies the following estimation:

Lemma 3.2.3 Let u be a H¨older continuous function in a subset Ω of Cn Then

Our second main result is the following theorem

Theorem 3.2.4 Let Ω be a strictly pseudoconvex domain and let G, u, v, φ be as

Conclusion and recommendations

I Conclusion

continuity of the PerronBremermann envelope of plurisubharmonic functions inbounded hyperconvex domains The thesis achieves the following main results

• Prove the existence of a solution to the Dirichlet problem for the complex

• Prove the solution to the problem MA(Ω, f, φ) isn’t continuous in the usual

-regular domian

• Prove the problem Dirichlet M(Ω, µ) when Ω is a strictly pseudoconvex main with a global solution if the problem has a local solution

of plurisubharmonic functions in bounded hyperconvex domains

• Research the Hessian equation for plurifinely open set domains.Finally, we would like to respectfully receive valuable suggestions from read-ers on research directions and new issues related to the thesis topic to continuedeveloping this research direction.

PUBLISHED WORKS RELATED TO THE DISSERTATION TITLE

1 N.X Hong and P.T Lieu, The Dirichlet Problem for the complex Monge Ampere operator on strictly pluri nely pseudoconvex domains, Complex Anal ysis and Operator Theory., (2021) 15:124

2 N.X Hong and P.T Lieu, Local Holder continuity of solutions of the complex Monge-Ampere equation, Journal of Mathematical Analysis and Applications, 507 (2022), 125737

3 T.V Long, N.X Hong and P.T Lieu, Continuity of the Perron-Bremermann envelope of plurisubharmonic functions, Collectanea Mathematica,

https://doi.org/10.1007/s13348-023-00405-9