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Zeitschrift für Analysis und ihre Anwendungen


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Volume 2, Issue 6, 1983, pp. 511–521
DOI: 10.4171/ZAA/82

Published online: 1983-12-31

Uniform approximation by solutions of general boundary value problems for elliptic equations of arbitrary order I

Günther Wildenhain[1]

(1) Universität Rostock, Germany

Let $\Omega \subset \mathbb R^n$ be a bounded, smooth domain, $\Gamma$ a closed, smooth, $(n-1)$-dimensional surface in the interior of $\Omega$ and $V$ an open subset of the boundary $\partial \Omega$. In $\Omega$ we consider a properly elliptic differential operator $L$ of arbitrary order with smooth coefficients. Let $B_1, \dots, B_m$ be a normal system of boundary operators on $\partial \Omega$, which fulfils the classical roots condition. $L_V(\Gamma)$ denotes the space of the restrictions on $\Gamma$ of the functions from $$L_V(\Omega) = \{u: Lu = 0 \: \mathrm {in} \: \Omega, B_1 u|_{\partial \Omega} =\dots = B_m u|_{\partial \Omega} = 0 \: \mathrm {in} \: \partial \Omega \setminus V \}.$$ Among other things it is proved, that the space $L_V(\Gamma)$ is dense in the space $W^{m-1} (\Gamma)$ of the Whitney-Taylorfields of the order $m-1$, i.e. all derivatives up to the order $m-1$ can be uniformly approximated on $\Gamma$.

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Wildenhain Günther: Uniform approximation by solutions of general boundary value problems for elliptic equations of arbitrary order I. Z. Anal. Anwend. 2 (1983), 511-521. doi: 10.4171/ZAA/82