Một số vấn đề trong lý thuyết đa thế vị
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Một số vấn đề trong lý thuyết đa thế vị
VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFDOCTOR OF PHILOSOPHY IN MATHEMATICSHANOI - 2021https://khothu vien .comVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAT DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYSpeciality: Mathematical AnalysisSpeciality code: 9460102 (62 46 Một số vấn đề trong lý thuyết đa thế vị 01 02)DISSERTATIONSUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DECREE OFDOCTOR OF PHILOSOPHY IN MATHEMATICSSupervisor: Prof. Dr.Sc. PTĨMột số vấn đề trong lý thuyết đa thế vị
AM HOANG ĨIĨLPProf. Dr.Sc. DĨNIĨ TĨICN CUONGHANOI - 2021IntroductionIt lias been known since the 19th century that gravity and electrostatic forces arVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị tion. The term “(classical) potential theory” arose to describe a linear theory associated to the Laplacian. This theory focused on harmonic functions, subharmonic functions, the Dirichlet problem, harmonic measure, Green’s functions, potentials and capacity in several real variables.The potential t Một số vấn đề trong lý thuyết đa thế vị heory in two dimensional space, which is always considered as the potential theory in the complex plane, has attracted considerable interest since itMột số vấn đề trong lý thuyết đa thế vị
is closely related to complex analysis. In particular, there is a connection between Laplace’s equation and analytic functions. While the real and imaVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị eal part of an analytic function. In general, some techniques of complex analysis, particularly conformal mapping, can be used to simplify proofs of some results in the potential theory, while some theorems in the potential theory have analogies and applications in complex analysis.In the 20 11 cent Một số vấn đề trong lý thuyết đa thế vị ury, pluripotcntial theory was developed as the complex multivariate analogue of the classical potential theory in the complex plane. This theory is hMột số vấn đề trong lý thuyết đa thế vị
ighly non-linear and associated to complex Mongc-Ampcrc operators. The basic objects arc plurisubharmonic functions of several complex variables that VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị able. The plurisubharmonic functions are also considered as subharmonic functions on several real variables which are invariant with respect to all biholoniorphic coordinate systems.In this dissertation, we study some specific problems in the plnripotential theory and the potential theory.viiIn Chap Một số vấn đề trong lý thuyết đa thế vị ter 1, motivated by the fact that two subharmonic functions which agree almost everywhere on a domain with respect to Lebesgue measure must coincide eMột số vấn đề trong lý thuyết đa thế vị
verywhere on that domain, we arc interested in the following problem.Problem 1. Whether we can conclude that two subharmonic functions on a domain ofRVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị nifold?Chapter 1 is devoted to answer Problem 1 completely. For this purpose, we prove two main theorems with similar assumptions. They concern restrictions of subharmonic functions in Q to a Borel subset A’ c Q which, together with a measure /z. is subject to some technical assumptions. These allow Một số vấn đề trong lý thuyết đa thế vị K to have co-dimension one (and a little more, but not two), with // being more or less a corresponding Hausdorff measure. The first main result (TheMột số vấn đề trong lý thuyết đa thế vị
orem 1.3.3) is an extension of the mean value theorem. It states that the mean value theorem in an infinitesimal form still holds when restricted to AVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị n an upper semicontinuous function and a subharmonic function which holds almost everywhere (with respect to /z) on A' actually holds at every point of K. By these theorems, we prove that Problem 1 has a positive answer in the case of hypersurfaces. We also provide a counterexample (Example 1.4.4) i Một số vấn đề trong lý thuyết đa thế vị n the case of subspaces of higher co-dimension. In addition, we apply the main theorems to Ahlfors-David regular sets to obtain some consequences, andMột số vấn đề trong lý thuyết đa thế vị
prove other versions of the main results in terms of measure densities.In Chapter 2, we study the Dirichlet problem for the complex Monge-Ampere equaVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL F Một số vấn đề trong lý thuyết đa thế vị ere equation is continuous outside an analytic set but u may not be continuous in Í2.This problem arises from the fact that there are some plurisubharmonic functions which arc not continuous in the whole domain, though they are continuous outside an analytic set. For example, tz(z) = — (— log ||xrII Một số vấn đề trong lý thuyết đa thế vị )1/2 is not continuous in the whole unit ball IB2”, but it is continuous in B2"\{()}. In studying this problem, we prove a sufficient condition (TheorMột số vấn đề trong lý thuyết đa thế vị
em 2.4.8) which relaxes assumptions of a well-known result of Kolodziej (Theorem B in [26]) to some technical assumptions. These assumptions naturallyVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL FVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGYINSTITUTE OF MATHEMATICSDO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORYDISSERTATIONSUBMITTED IN PARTIAL FGọi ngay
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