I'm trying to prove that the integral $\int_{U}u^{4}$ is well-defined for $u\in H_{0}^{2}(U)$ and $U\subset\mathbb{R}^{n}$ open, bounded, $n\leq3$. In other words, I need to prove that $H_{0}^{2}(U)$ ($=W_{0}^{k,p}(D)$ with $k=p=2$) can be embedded in $L^{4}(U)$. Note that there are no smoothness assumptions on the boundary.
I tried to distinguish the cases $n=1,2,3$. For $n=1$ I think we can use Morrey's inequality (which requires $1=n<p=2$) to prove that $u$ is Hölder continuous (up to a set of measure zero). Thus $$H_{0}^{2}(U)\hookrightarrow C^{0}(\overline{U})\hookrightarrow L^{\infty}(U)\hookrightarrow L^{4}(U).$$
However, I cannot come up with the right arguments for the cases $n=2,3$. The only seemingly useful embedding-theorem I can find requires $n>kp=4$, which we do not have.
If $n=3$, then the Sobolev conjugate of $2$ is equal to $6$, and since $\nabla^2 u\in L^2(U)$, we have $\nabla u\in L^6(U)$. Hence, Morrey's theorem implies that $u\in L^{\infty}(U)$.
If $n=2$, then $\nabla u$ belongs to every $L^p(U)$ space, for any $p\geq 1$. In particular $\nabla u\in L^3(U)$, and Morrey's theorem shows that $u\in L^{\infty}(U)$.