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Zeitschrift für Analysis und ihre Anwendungen


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Volume 5, Issue 1, 1986, pp. 59–69
DOI: 10.4171/ZAA/180

Published online: 1986-02-28

Approximation by Solutions of Elliptic Equations

Uwe Hamann[1] and Günther Wildenhain[2]

(1) Universität Rostock, Germany
(2) Universität Rostock, Germany

Let $\Omega \subset \mathbb R^n$ be a bounded, smooth domain, $\Gamma$ a closed, smooth, $(n-1)$-dimensional surface with boundary 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 order $2m$ with smooth coefficients. Let $(B_1, \dots, B_m)$ be normal system of boundary operators on $\partial \Omega$, which fulfils the classical root condition. $L_V(\Gamma)$ denote the space of the restrictions on $\Gamma$ of the functions from $$L_V(\Omega) = \{u \in C^\infty \Omega \colon Lu = 0 \; \mathrm {in}\; \Omega, \; B_1u|_{\partial \Omega} = \cdots = B_mu|_{\partial \Omega} = 0 \; \mathrm {in} \; \partial \Omega \; \backslash \; V\}.$$ It is proved that $L_V(\Gamma)$ is dense in the space $W_p^{2m-1/p} (\Gamma) (p > 1)$.

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Hamann Uwe, Wildenhain Günther: Approximation by Solutions of Elliptic Equations. Z. Anal. Anwend. 5 (1986), 59-69. doi: 10.4171/ZAA/180