论文标题
关于拉普拉斯的特征值的界限
About Bounds for Eigenvalues of the Laplacian with Density
论文作者
论文摘要
令$ m $表示{\ mathbb n} $的dimension $ n \的紧凑，连接的riemannian歧管。我们假设$ m $具有平稳且连接的边界。用$ g $和$ {\ rm d} v_g $表示，$ m $上的riemannian公制和关联的卷元素。令$δ$为$ m $上的laplace运算符，配备了加权音量表单$ {\ rm d} m：= {\ rm e}^{ - h} { - h} \，{\ rm d} v_g $。我们对运算符$ l_h \ cdot感兴趣：= {\ rm e}^{ - h（α-1）}（Δ\ cdot +αg（\ nabla h，\ nabla h，\ nabla \ cdot））$，其中$ a> 1 $ and $α> 1 $ and $ h \ in C^2（m）$。本文的主要结果指出，如果边界是非空的，则具有加权拉普拉斯$ l_h $的特征值的上限具有Neumann边界条件。
Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated volume element. Let $Δ$ be the Laplace operator on $M$ equipped with the weighted volume form ${\rm d}m:= {\rm e}^{-h}\,{\rm d}v_g$. We are interested in the operator $L_h\cdot:={\rm e}^{-h(α-1)} (Δ\cdot +αg(\nabla h,\nabla\cdot))$, where $α> 1$ and $h\in C^2(M)$ are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian $L_h$ with the Neumann boundary condition if the boundary is non-empty.
