Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of $\mathbb R^{d}$
Volume 251 / 2020
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 the particular cases of $\Omega _+=\mathbb {R}^{d-1}\times (0,\infty )$ and $\Phi _{t}(u)=\exp (-tu)$, $u \geq 0$, $t \gt 0$, 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.
