【作者】 张涛；

【导师】 周春琴；

【作者基本信息】 上海交通大学 ， 基础数学， 2019， 博士

【摘要】 Liouville型方程在数学物理问题以及几何分析当中具有广泛的应用,比如平均场方程,预定Gauss曲率问题以及Chern-Simons理论等问题.在过去几十年,数学工作者对Liouville型方程的研究取得了一系列重要的成果,极大的促进了数学和物理的进展.在研究Liouville型方程时,一个重要的方面是爆破分析,因为它和解的收敛性质密切相关.通过爆破分析,可以处理Moser-Trudinger型不等式,局部一致估计,解的存在性等问题.对于具有Dirichlet边界条件的Liouville型方程,很多数学工作者研究了这类方程,取得了相当多的成果.但是近些年来,随着几何中的预定高斯曲率和边界曲率问题研究的需要,具有指数Neumann边界条件的Liouville型方程也引起了数学工作者的注意,并且对无奇性项的方程取得了一些成果.但是,如果内部或者边界具有奇点,对这类具有奇性项和具有指数Neumann边界条件的Liouville型方程的研究是具有重要意义的.基于此,我们将研究这类具有奇性项的方程.本文的具体内容可概括如下:第一章主要介绍本文的研究背景以及具有指数Neumann边界条件的Liouville型方程的一些研究成果,同时介绍本文的主要研究内容.第二章主要研究无奇性项的具有指数Neumann边界条件的Liouville型方程在爆破点处的能量量子性态.我们将借助于“sup+inf”不等式建立Li-Shafrir型的能量量子性态结果.第三章主要研究具有奇性项和指数Neumann边界条件的Liouville型方程的爆破分析及在奇性爆破点处的能量量子性态.当方程具有奇性项时,它的研究将更为复杂.我们将借助于Tarantello型的分解引理和新的“sup+inf”不等式对这类方程进行更细致的研究.第四章主要研究具有奇性项和指数Neumann边界条件的Liouville型方程在奇性爆破点处的局部一致估计.我们将分别讨论α(?)N和α ∈N+两种情形,建立了两种不同的局部一致估计.更多还原

【Abstract】 The Liouville type equation has been widely used in mathematical physics problems and geometric analysis,for instance,the problem of mean field equation,prescribing Gaussian curvature and Chern-Simons theory,etc.In the past few decades,mathematic researchers have made many important achievements in the research of Liouville type equation,which greatly promoted the progress of mathematics and physics.For the research of the Liouville type equation,an important aspect is blow-up analysis,since it is related to the the convergence of sequence of solutions.By the trick of blow-up analysis,we can treat Moser-Trudinger equality,the local uniform estimate and problems of existence of solution,etc.For the Liouville type equation with Dirichlet boundary condition,many researchers studied this type equation and made a lot of achievements.But in recent years,arising in prescribing Gaussian curvature and boundary curvature problem,the Liouville type equation under exponential Neumann boundary condition has also attracted the attention of mathematic researchers and have some progresses for this type equation without singular data.But when interior or the boundary has singularity,the research of Liouville type equation with exponential Neumann boundary condition and with singular data has important significance.On the base,we mainly analyze this type equation.The specific content of this article can be summarized as follows.In the first chapter,the research background of this paper and the research progress of the Liouville type equation with exponential Neumann boundary condition are briefly introduced,and the main contents of this paper are also introducedIn the second chapter,we introduce the quantization property of the blow up value for the Liouville type equation with exponential Neumann boundary condition without singular data.We will use the "sup+inf" inequality to establish Li-Shafrir’s type energy quantization resultIn the third chapter,we introduce the blow up analysis for Liouville type equation with exponential Neumann boundary condition and with singular data And we give the quantization property of the blow up value for the singular blow up point.When the equation has singular data,the study of the equations will be more difficult.We give the version of Tarantello’s decomposition Lemma and new "sup+inf" type inequality under exponential Neumann boundary conditions separately.In the fourth chapter,We establish uniform estimates for blow-up sequence of solutions near an isolated singular blow up point.We treat two cases α(?)N andα∈N+ separately,and obtain two different uniform estimates.更多还原