On annihilators of harmonic vector fields

Abstract

Let Ω⊂ℝ^N be a smooth bounded domain. We characterize smooth vector fields g on ∂Ω which annihilate all harmonic vector fields f in Ω continuous up to ∂Ω, with respect to the pairing$\left\langle {f,g} \right\rangle = \int\limits_{\partial \Omega } {f \cdot gd\sigma }$ (dσ denotes the hypersurface measure on ∂Ω). In addition, we extend these results to differential forms with harmonic vector fields being replaced by harmonic fields, i.e., forms f satisfying df=0, δf=0. A smooth form g on ∂Ω is an annihilator if and only if suitable extensions, u and v, into Ω of its normal and tangential components on ∂Ω, satisfy the generalized Cauchy-Riemann system du=δv, δu=0, dv=0 in Ω. Finally, we prove that the described smooth annihilators are weak^* dense among all annihilators. Bibliography: 12 titles.