坍缩区域的特征值与漂移拉普拉斯算子
Eigenvalues of collapsing domains and drift Laplacians
摘要：
通过在拉普拉斯算子中引入一个权函数，Bakry和Emery定义了“漂移拉普拉斯算子”来研究扩散过程。我们的第一个主要结果是，给定一个Bakry-Émery流形，存在一个自然关联的图族，当图坍缩到流形时，其特征值收敛到漂移拉普拉斯算子的特征值。这个结果的应用包括$\R^n$中域的Dirichlet特征值和$\R^{n+1}$中域的Neumann特征值之间的一个新的关系和一个新的最大值原理。利用我们的主要结果和最大值原理，我们推广了\emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Émery manifolds} .
英文摘要：
By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Émery manifolds}.
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