On annihilators of harmonic vector fields

B. Gustafsson ; D. Khavinson

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 90-108

Résumé

For $\Omega\subset\mathbb R^n$ a smoothly bounded domain we characterize smooth vector fields $g$ on $\partial\Omega$ which annihilate all harmonic vector fields $f$ in $\Omega$ continuous up to $\partial\Omega$, with respect to the pairing $\langle f,g\rangle=\int_{\partial\Omega}f\cdot g\,d\sigma$ ($d\sigma$ denotes the hypersurface measure on $\partial\Omega$). Also, we extend these results to the context of differential forms with harmonic vector fields being replaced by harmonic fields, i.e., forms $f$ satisfying $df=0$, $\delta f=0$. Then a smooth form $g$ on $\partial\Omega$ is an annihilator if and only if suitable extensions, $u$ and $v$, into $\Omega$ of its normal and tangential components on $\partial\Omega$ satisfy the generalized Cauchy–Riemann system $du=\delta v$, $\delta u=0$, $dv=0$ in $\Omega$. Finally we prove that the smooth annihilators we describe are weak$^*$ dense among all annihilators. Bibl. 12 titles.