Question on Gagliardo-Nirenberg.

Question

On page 679 of this paper, the authors claim they can get a special case of Gagliardo-Nirenberg with a constant of 1/2. They prove this using functions in $C_0^\infty(\mathbb{R}^2)$, for which the proof in the footnote is clear. I have no idea how they can extend this to functions in $H^1(\Omega)$ without picking up a constant from an extension operator. If you have the time, this small part is very confusing to me.

Answer 1

They don't deal with functions in $H^1(\Omega)$. Recall that $u$ vanishes on the boundary of $\Omega$ (conditions (1)). Since the boundary is smooth, this means $u\in H^1_0(\Omega)$. Therefore, the zero extension of $u$ to $\mathbb R^n$ is an $H^1$ function. The extension by zero does not increase the $H^1$ norm.

Since a compactly supported $H^1$ function can be approximated in $H^1$ norm by compactly supported $C^\infty $ functions, the inequality follows.