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ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM
Published online by Cambridge University Press: 20 August 2012
Abstract
Let $\varphi _0$ and $\varphi _1$ be regular functions on the boundary $\partial D$ of the unit disk $D$ in $\mathbb {R}^2$, such that $\int _{0}^{2\pi }\varphi _1\,d\theta =0$ and $\int _{0}^{2\pi }\sin \theta (\varphi _1-\varphi _0)\,d\theta =0$. It is proved that there exist a linear second-order uniformly elliptic operator $L$ in divergence form with bounded measurable coefficients and a function $u$ in $W^{1,p}(D)$, $1 \lt p \lt 2$, such that $Lu=0$ in $D$ and with $u|_{\partial D}= \varphi _0$ and the conormal derivative $\partial u/\partial N|_{\partial D}=\varphi _1$.
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- Research Article
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- Copyright © 2012 Australian Mathematical Publishing Association Inc.