论文标题
关于DIRICHLET EIGENVALUE问题和共形Skorokhod嵌入问题
On the Dirichlet eigenvalue problem and the conformal Skorokhod embedding problem
论文作者
论文摘要
在GROSS的最新工作中，陈述和解决了以下问题：鉴于有限第二刻的量度$μ$，找到一个简单连接的域$ u $ in $ \ cc $中的$ u $，使得在$ u $分布时，布朗尼运动的实际部分停止了，为$μ$。 Gross开发的结构产生的域相对于真实轴是对称的，但其他作者已经指出，其他域也可能是可能的，特别是有许多示例具有垂直射线始于域中的一个属性，完全位于域内。在本文中，我们为Gross提出的问题提供了一个新的解决方案，并表明这些其他情况是此方法的特殊情况。我们进一步表明，该方法生成的域具有其属性，该属性始终具有与固定分布$μ$相对应的所有可能域中的laplacian操作员频谱的最低率（按照拉普拉斯运算符的频谱），这为Mariano和Panzo提出的问题提供了部分解决方案。我们证明该域是独特的，只要强加了某些条件，并以此为例。我们还描述了一种识别域边界曲线的方法，并讨论其他几个相关主题。
In a recent work by Gross, the following problem was stated and solved: given a measure $μ$ with finite second moment, find a simply connected domain $U$ in $\CC$ such that the real part of a Brownian motion stopped when it leaves $U$ is distributed as $μ$. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution $μ$, which gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.
