论文标题
Laplacian的有限反射组和对称扩展
Finite reflection groups and symmetric extensions of Laplacian
论文作者
论文摘要
令$ w $是与root System $ r $ in $ \ Mathbb r^d $相关的有限反射组。令$ c _+$表示阳性韦尔室。考虑$ \ Mathbb r^d $的开放子集$ω$，相对于$ w $的反射。令$ω_+=ω\ cap c _+$为$ω$的正部分。我们定义了一个家庭$ \ { - δ_η^+\} $的laplacian $-Δ__{ω_+} $的自相关扩展，由同型同粒子标记为$η\ colon w \ to \ colon w \ to \ colon w \ to \ to \ {1，-1，-1 \} $。在构建这些$η$ -laplacians $η$ - $ symememations在$ω$上的构成。包括neumann laplacian $-Δ_{n，ω_+} $，对应于$η\ equiv1 $。如果包括$ h^{1}（ω）= h^{1} _0（ω）$，则包括dirichlet laplacian $-Δ__{d，ω_+} $，并且对应于$η= {\η= {\ rm sgn} $;否则，分别考虑了Dirichlet Laplacian。 Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$.我们证明了这些对运营商的积分内核之间的关系，这些内核是根据$ w $支配的对称性给出的。
Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $Ω_+=Ω\cap C_+$ be the positive part of $Ω$. We define a family $\{-Δ_η^+\}$ of self-adjoint extensions of the Laplacian $-Δ_{Ω_+}$, labeled by homomorphisms $η\colon W\to \{1,-1\}$. In the construction of these $η$-Laplacians $η$-symmetrization of functions on $Ω$ is involved. The Neumann Laplacian $-Δ_{N,Ω_+}$ is included and corresponds to $η\equiv1$. If $H^{1}(Ω)=H^{1}_0(Ω)$, then the Dirichlet Laplacian $-Δ_{D,Ω_+}$ is either included and corresponds to $η={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.
