Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of $\mathbb{R}^d$

Abstract

For an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to a hyperplane and with positive part $Ω_+$, we consider the Neumann/Dirichlet Laplacians $-Δ_{N/D,Ω}$ and $-Δ_{N/D,Ω_+}$. Given a Borel function $Φ$ on $[0,\infty)$ we apply the spectral functional calculus and consider the pairs of operators $Φ(-Δ_{N,Ω})$ and $Φ(-Δ_{N,Ω_+})$, or $Φ(-Δ_{D,Ω})$ and $Φ(-Δ_{D,Ω_+})$. We prove relations between the integral kernels for the operators in these pairs, which in particular cases of $Ω_+=\mathbb{R}^{d-1}\times(0,\infty)$ and $Φ_{t}(u)=\exp(-tu)$, $u \geq 0$, $t>0$, were known as reflection principles for the Neumann/Dirichlet heat kernels. These relations are then generalized to the context of symmetry with respect to a finite number of mutually orthogonal hyperplanes.