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ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 1 / August 2013
- Published online by Cambridge University Press:
- 20 August 2012, pp. 51-55
- Print publication:
- August 2013
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- Article
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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 
$\partial u/\partial N|_{\partial D}=\varphi _1$.
