Extension of functions from Sobolev spaces

V. G. Maz'ya

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 231-236

Résumé

By definition, the domain $\Omega\subset\mathbb R^n$ belongs to the class $EW_p^l$ if there exists a continuous linear extension operator $W_p^l(\Omega)\to W_p^l(\mathbb R^n)$. An example is given of a domain $\Omega\subset\mathbb R^2$ with compact closure and Jordan boundary, having the following properties: (1) The curve $\partial\Omega$ is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) $\Omega\in EW_p^1$ for $p2$ and $\Omega\not\in EW_p^1$ for $p\ge2$. (3) $\mathbb R^2\setminus\overline\Omega\in EW_p^1$ for $p>2$ and $\mathbb R^2\setminus\overline\Omega\not\in EW_p^1$ for $p\le2$.