Limit in $W_{p} ^2$ implies limite in $L^{q}$.

Question

Let $u_{m}\in W_{p} ^{2} \cap W_{q} ^{2}$ be such that $u_{m} \rightarrow u$ in $W_{p} ^2$. Note that the domain $\Omega$ is a polygon. I want to show $$u_{m} \rightarrow u \text{ in } L^{q}$$. My textbook says this is because of Sobolev imbedding.

Answer 1

Let $\Omega \subset \mathbb R^2$ be a (bounded) polygon. In particular, $\Omega$ possesses a Lipschitz boundary. For all $p >1 $, you have that $W^{2,p}(\Omega)$ is continuously embedded (by the Sobolev embedding theorem) in $L^\infty(\Omega)$. That is, there is $C > 0$ with $$ \|u - u_m \|_{L^\infty(\Omega)} \le C \, \| u - u_m \|_{W^{2,p}(\Omega)} \to 0.$$