【作者】 石飞林；

【导师】 戴求亿；

【作者基本信息】 湖南师范大学 ， 应用数学， 2014， 博士

【摘要】 本文主要研究偏微分方程特征值和解的性态.偏微分方程来自于化学、物理和生物等科学领域,具有十分强烈的实际背景.它与数学中的其他很多分支密切相关,如微分几何,复分析,调和分析等,逐渐成为数学学科中最重要的研究领域之一.特征值与解的性态问题是偏微分方程中的基本问题.因此,对于偏微分方程中的特征值和解的性态的研究不但具有科学意义,而且具有潜在的应用价值.本文的主要内容安排如下：第1章简述有关偏微分方程特征值和解的性态的研究背景及本文的主要工作.第2章是预备知识.介绍了本文要用到的数学术语和数学工具.第3章,我们讨论如下Laplacian Robin特征值问题其中Ω∈Rπ(n≥2)是有界区域,αΩ是Lipschitz的,ν为边界上的单位外法向量以及0<β<+∞.本章的结论主要有两个：一个我们利用schwarz对称化方法和函数重排技巧得到了问题(1)的前两个特征值之比的一个上界估计.该估计在某个特定的限制条件之下是M.S.Ash-baugh和R.D.Benguria[4,6]证明的PPW猜测的一个推广；另一个是证明了问题(1)的第一特征函数满足Chiti型反向Holder不等式.在第4章中,我们研究如下k-Hessian算子的特征值问题其中Ω是Rπ中光滑有界凸区域.我们推导出问题(2)的主特征值的变分公式并推导出相关的超定问题；其次,我们得到了所推导的超定问题的Serrin型对称性结果.更多还原

【Abstract】 In this thesis, we study eigenvalues and the properties of solution for par-tial differential equations. Partial differential equations involved in a number of issues from the chemistry, physics and mathematical model of the biological field, with a strong practical background. It has become one of the most im-portant fields of study in mathematical, which interacts with many branches of mathematics, such as differential geometry, complex analysis and harmonic analysis. Eigenvalues and the properties of solution are basic problems in partial differential equations. Therefore, the study on eigenvalues and the properties of solution for partial differential equations has scientific significance and potential applications. The main contents are organized as follows:In chapter1, we state the background of eigenvalues and the properties of solution for partial differential equations and the main work of this article.In chapter2, we introduce some mathematical terms and tools as prelimi-naries that will be used in the following chapters.In chapter3, Let Ω (?) Rn(n≧2) be a bounded domain with boundary (?)Ω, ν be the outward unit vector normal to (?)Ω, and0<β<+∞be a parameter. We discuss the following Robin eigenvalue problem The aim of this chapter is twofold. One is an upper bound for the ratio of the first two eigenvalues by Schwarz symmetrization method and rearrange-ment techniques, which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria in [4] and [6], the other is a Chiti type reverse Holder inequality for the first eigenfunction.In chapter4, we concern the following eigenvalue problem of k-Hessian operator where Ω is a smooth bounded convex domain in Rn. First we devote to deduce the first variational formula and some related overdetermined problems for the principle eigenvalue of problem (2), and then prove Serrin type symmetry result for our overdeter-mined problems.更多还原