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

Abstract

For an open subset $\Omega$ of ${\mathbb{R}}^d$, symmetric with respect to a hyperplane and with positive part ${\Omega }_{+}$, we consider the Neumann/Dirichlet Laplacians $-{\Delta }_{N/D,\Omega }$ and $-{\Delta }_{N/D,{\Omega }_{+}}$. Given a Borel function $\Phi$ on $[0,\infty )$ we apply the spectral functional calculus and consider the pairs of operators $\Phi (-{\Delta }_{N,\Omega })$ and $\Phi (-{\Delta }_{N,{\Omega }_{+}})$, or $\Phi (-{\Delta }_{D,\Omega })$ and $\Phi (-{\Delta }_{D,{\Omega }_{+}})$. We prove relations between the integral kernels for the operators in these pairs, which in particular cases of ${\Omega }_{+}={\mathbb{R}}^{d-1}\times (0,\infty )$ and ${\Phi }_t(u)=\exp (-tu)$, $u\ge 0$, $t>0$, 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.

25 pages