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Zapiski Nauchnykh Seminarov POMI, 1996, Volume 232, Pages 90–108
(Mi znsl3678)
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This article is cited in 8 scientific papers (total in 8 papers)
On annihilators of harmonic vector fields
B. Gustafssona, D. Khavinsonb a Matematiska institutionen, Kungl. Tekniska Högskolan, Sweden
b Department of Mathematical Sciences, University of Arkansas, USA
Abstract:
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.
Received: 20.08.1995
Citation:
B. Gustafsson, D. Khavinson, “On annihilators of harmonic vector fields”, Investigations on linear operators and function theory. Part 24, Zap. Nauchn. Sem. POMI, 232, POMI, St. Petersburg, 1996, 90–108; J. Math. Sci. (New York), 92:1 (1998), 3600–3612
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https://www.mathnet.ru/eng/znsl3678 https://www.mathnet.ru/eng/znsl/v232/p90
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