Suppose $\Omega$ is bounded. Then, is it true that $W_0^{1,q}(\Omega)\subset W_0^{1,p}(\Omega)$ whenever $p\leq q$? It seems like it should certainly be true since we do know that $L^q(\Omega)\subset L^p(\Omega)$ under the same hypothesis. But I'm wondering if I'm missing a subtlety.
Asked
Active
Viewed 123 times
0
-
Are you defining Sobolev spaces by the formulation involving integrating weak derivatives? – πr8 May 29 '17 at 13:42
-
1The inclusion is bounded on $C^\infty_0(\Omega)$ because $p \le q$ and due to the boundedness of $\Omega$. If follows by the usual density argument for Sobolev spaces. – Hans Engler May 29 '17 at 14:29