I'm stuck proving a Gagliardo-Nirenberg Interpolation-type inequality.
Typically authors prove the inequality for functions of their favorite regularity and try a density argument. This often requires them to approximate a Sobolev function with a sequence that converges to it in several different Sobolev norms,and I don't know how to guarantee the existence of such a sequence.
In particular, I'm looking for a reason why a function $f$ in $H^2(\Omega)\cap L^\infty(\Omega)$ can be approximated by a single sequence $u_n$ converging in $H^2$ and $L^\infty$. Here $\Omega$ is a bounded region in $\mathbb{R}^2.$
If $\Omega \subset \mathbb R^2$, then $H^2(\Omega)$ is continuously embedded in $L^\infty(\Omega)$. Hence, the convergence in $H^2(\Omega)$ implies the convergence in $L^\infty(\Omega)$.
As pointed out by Quickbeam2k1 in the comments, one can not approximate a function $f$ from $H^2(\Omega) \cap L^\infty(\Omega)$ in the $L^\infty(\Omega)$-norm by continuous functions if the space dimension is large enough ($n \ge 4$). The reason is that $f$ might be discontinous, but every $L^\infty(\Omega)$-limit of continuous functions is continuous.