A Note on Some Non-Local Boundary Conditions and their Use in Connection with Beltrami Fields

Abstract

We consider two operators $A_{0},B_{0}$ between two Hilbert spaces satisfying $A_{0}\subseteq-B_{0}^{\ast}$ and $B_{0}\subseteq-A_{0}^{\ast}$ and inspect extensions $A^{\#}$ and $B^{\#}$ of $A_{0}$ and $B_{0}$, respectively, whose domain consists of those elements satisfying an abstract periodic boundary condition. The motivating example is the derivative on some interval, where the so-defined realisation gives the classical derivative with periodic boundary conditions. We derive necessary and sufficient conditions for the operator equality $A^{\#}=-\left(B^{\#}\right)^{\ast}$ and illustrate our findings by applications to the classical vector analytic operators $\mathrm{grad},\,\mathrm{div}$ and $\mathrm{curl}$. In particular, the realisation $\mathrm{curl}^{\#}$ naturally arises in the study of so-called Beltrami fields.