0

Theorem (Sobolev embedding) Let $1<p\leq q<\infty,\, s\geq 0$ and $\Omega$ an open set of $\mathbb{R}^n$. Then if $s<np$, then for any $q\in [p, \frac{np}{n-sp}]$, \begin{align} H^{s,p}(\Omega)\hookrightarrow L^q(\Omega) \end{align}

Question. Is the embedding also valid if we change $\Omega$ to $\overline{\Omega}$?

eraldcoil
  • 3,888
  • If you replace $\Omega$ by $\bar \Omega$ what should change? $\bar\Omega \setminus \Omega$ has zero measure in many cases. – daw Nov 04 '24 at 20:39
  • Oh! I see.. I read https://math.stackexchange.com/questions/613408/when-does-the-boundary-have-measure-zero. Therefore, if $\Omega$ is open and convex, then we are done because $\overline{\Omega}\setminus \Omega$ has zero measure. Thanks. – eraldcoil Nov 04 '24 at 20:45

0 Answers0