A completeness theorem on the boundary in Besov spaces

Source abstract

Abstract Let { ω k } {\{\omega_{k}\}} be a basis of polynomial solutions to the equation E ⁢ u = 0 {Eu=0} , where E is a higher-order elliptic operator with real constant coefficients and no lower-order terms, in any number of variables. The aim of the paper is to prove that the system { ( ω k , ∂ ν ⁡ ω k , … , ∂ ν m - 1 ⁡ ω k ) } {\{(\omega_{k},\partial_{\nu}\omega_{k},\dots,\partial^{m-1}_{\nu}\omega_{k})\}} is complete in the Besov space [ B p , p 1 - 1 / p ⁢ ( ∂ ⁡ Ω ) ] m {[B^{1-{1}/{p}}_{p,p}(\partial\Omega)]^{m}} ( 1 ≤ p < ∞ ) {(1\leq p<\infty)} , Ω being a C ∞ {C^{\infty}} bounded domain in ℝ n {\mathbb{R}^{n}} such that ℝ n ∖ Ω ¯ {\mathbb{R}^{n}\setminus\overline{\Omega}} is connected.