Finite reflection groups and symmetric extensions of Laplacian

TL;DR

This paper constructs and analyzes a family of symmetric extensions of the Laplacian on domains symmetric under finite reflection groups, revealing relations between their integral kernels influenced by group symmetries.

Contribution

It introduces a new family of self-adjoint Laplacian extensions parametrized by group homomorphisms, connecting classical boundary conditions with symmetry considerations.

Findings

Established relations between integral kernels of different Laplacian extensions.

Unified treatment of Neumann and Dirichlet Laplacians within the symmetry framework.

Demonstrated how group symmetries influence spectral properties of Laplacian extensions.

Abstract

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 $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\Omega_+=\Omega\cap C_+$ be the positive part of $\Omega$. We define a family $\{-\Delta_{\eta}^+\}$ of self-adjoint extensions of the Laplacian $-\Delta_{\Omega_+}$, labeled by homomorphisms $\eta\colon W\to \{1,-1\}$. In the construction of these $\eta$-Laplacians $\eta$-symmetrization of functions on $\Omega$ is involved. The Neumann Laplacian $-\Delta_{N,\Omega_+}$ is included and corresponds to $\eta\equiv1$. If $H^{1}(\Omega)=H^{1}_0(\Omega)$, then the Dirichlet Laplacian $-\Delta_{D,\Omega_+}$ is either included and corresponds to $\eta={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\Psi(-\Delta_{N,\Omega})$ and $\Psi(-\Delta_{\eta}^+)$, or $\Psi(-\Delta_{D,\Omega})$ and $\Psi(-\Delta_{D,\Omega_+})$, where $\Psi$ 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 $W$.